text:
The Unicyclopedia/Ultimate wheel

The Ultimate Wheel is a simpler form of the unicycle, consisting of a wheel with two pedals, but with no seat or frame, the lateral stability of the wheel is the most crucial skill involved in learning to ride. Large leg pads much like those worn by a hockey goalie can be worn in the learning stage.

Mounting

 * Freemount
 * This is done pretty much the same as a normal unicycle. Place the UW with the pedal corresponding to your weaker foot at its lowest point. Put your weak foot on the pedal and lean the wheel toward your dominant foot. This stops the wheel from flipping when you lift up your dominant foot. As soon as both feet are on the pedals, apply pressure and head off.

 * Roll Mount
 * The hardest thing here is to get the wheel to roll in a straight line. As you experiment, this improves. After it is rolling, jog next to it until the pedal that corresponds to your weaker foot is at the lowest point. Now, put your weak foot on that pedal followed immediately by the dominant foot. When the wheel is rolling, it is oddly stable. Here's a clip that shows it. It is also possible to do this trick in reverse where the wheel is rolling towards you. You can see it at 1:04 of this clip

 * Kickup Mount
 * This skill is much easier on an UW than on a unicycle. Firstly, lay the wheel on the ground, placing your dominant foot on the upwards-facing pedal, and place the toes of your weaker foot under the rim of the wheel by the downwards facing pedal. Lift the wheel with your weaker foot while at the same time easing your weight on the upwards facing pedal. Your dominant foot should now be at its lowest point while the weaker foot should be seated nicely on the pedal at its highest point. This position is similar to the normal freemount - just reversed.

 * Jump Mount
 * This is very similar to a jump mount on a unicycle, except more difficult to master. Firstly, hold the UW in a position where the pedals are aligned horizontally. From here, jump onto the UW without affecting the angle of the pedals. If you aim correctly, you will land on the pedals. You should now push down on the forward pedal using your dominant foot, and ride away.

 * Spin Mount
 * This skill is similar to the jump mount, except you spin the wheel before jumping. Start learning the trick by spinning the wheel 180 degrees and work your way up. The pedals should be aligned vertically before you spin. You can see it at 0:23 of this clip

 * Air Mount
 * With this skill you jump in the air while holding the Ultimate Wheel and mount it before you touch the ground. You can see it at 1:14 of this clip

Riding

 * Forward
 * This is just regular riding which in itself is extremely hard. Here's a clip that shows it

 * Idling
 * This is rocking back and forth (with your pedals in horizontal position) and staying at one place. You can see it at 4:00 of this clip

 * Backward
 * Here you do the same as forward except you pedal backwards. You can see it at 1:32 of this clip

Tricks

 * 180 Spin
 * While riding the UW twist your body 180 degrees and continue riding in reverse. You can see it at 1:53 of this clip

 * Freewheel Jump
 * Start off by riding your Ultimate Wheel in a circle. Jump off the wheel so that it continues rolling in a circle on it's own. When it comes back around jump on it (basically a roll mount) and keep riding. Once you master this technique, you can add 180 or 360 spins as you jump on the wheel or even do everything in reverse. You can see it at 2:08 of this clip.

 * Double Freewheel Jumps
 * This is the same trick but with two UW's. Start off by riding your wheel in a circle with a second wheel in your hand. As your riding, roll the wheel in your hand so it keeps going in a circle. Now wait till the other wheel rolls about half a circle and jump of your wheel. The wheel that was in your hand should now be coming back to you so you can jump back onto it. Practice and timing is everything. You can see it at 2:38 of this clip.

 * Twisty Jump
 * Start off by riding your ultimate wheel in a straight line. Jump up off the wheel, do a 180 degree spin and land back on the wheel (make sure your pedals are horizontal). Now continue riding in reverse. Note: Be careful with this trick as you can badly sprain or break an ankle. You can see it at 3:38 of this clip.

 * Leap of Faith
 * Start off by riding your ultimate wheel while holding another wheel in your hand. Now roll the wheel in your hand in a straight line and follow behind it. Jump from the wheel your riding onto the one you rolled. It's important to watch the pedals on the wheel as it rolls. You have to time it just right so that you land on the wheel right before the pedals line up horizontally. It may help to be slightly to the left or right of the wheel your jumping on instead of right behind it. You can see it at 3:18 of this clip.

 * Ultimate Spin
 * This feels exactly the same as a unicycle spin except that your on an ultimate wheel. You can see it at 4:28 of this clip.

 * Jump Rope
 * Start by idling with your pedals in the vertical position. Now hook your foot under the top pedal and start bouncing up and down. You can see it at 5:05 of this clip.

title: The Unicyclopedia/Ultimate wheel


text:
The Unicyclopedia/BC wheel

Introduction
The BC wheel (or Impossible Wheel) is a single wheel with pegs or small platforms connected to the axle. To ride you must balance on the pegs as the wheel is rolling. The name comes from the popular comic, B.C. by Johnny Hart, in which the characters travel by rolling on large stone wheels.

Riding the BC requires some different skills to the unicycle or Ultimate Wheel since fine control of the wheel's rotation is not possible. Since it requires very good balance it is not recommended for beginners. Advanced riders are able to do many tricks, including a variety of mounts, backwards riding, jumps and grinds.

In this video Leo Vandewoestijne shows how to ride a BC wheel

''	The reason I love to BC so much is this: the most fun I've ever had on a uni was when I first got one, and just learning to ride those first few days. I fell so much but I was progressing I never had a balance point it was always hit or miss, and I get that everyday with the BC wheel. There is never a balance point you can get down to minimal movement but your still focused solid on what you're doing.

~Evan Byrne''

Protective Gear
A very thick, strong, flexible, and heat resistant wrap or strap-on guard is important for braking and control. Gloves are recommended since, in some falls, you will hit the ground hands first. A skate-type helmet that protects the back of your neck is also recommended in case you fall backwards.

Mounting
Freemount: This is similar to the freemount for an ultimate wheel. Whilst standing still, place one foot on a plate and angle the wheel away from that leg. Angling the wheel prevents it from touching the leg when pressure is applied to the footplate. Roll the wheel forward and kick off from the ground with the other foot; this supplies the initial momentum for riding the wheel. The second foot must then be placed quickly onto the other plate to maintain the balance of the wheel.

In this video Leo Vandewoestijne shows the Freemount

Roll mount: The wheel must be rolled along the ground in a straight line, which requires some skill in itself. The rider jogs up behind the wheel until they are close enough to jump onto the footplates. At the point of jumping, the rider should be careful to be running at the same speed as the wheel or it will be very hard to land correctly. On landing, both feet should contact the plates at (as near as possible) the same time.

In this video Max shows the Roll Mount

Skate Mount This is a relatively easy skill. The rider puts one foot on a peg or plate and pushes like one would push a skateboard, then puts the other foot on the other plate and rides away. This mount requires some dexterity and skill with the wheel to keep the wheel from rubbing your leg and slowing down when you have your one foot on the plate/peg.

In this video Evan Byrne shows the Skate Mount

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: The Unicyclopedia/BC wheel


text:
The Unicyclopedia/Manufacturers

Axis
Basic but good quality beginners unicycles available in New Zealand and Australia.

Badger Bikes
Badger Bikes makes custom 20", 24", 26", 29" and 36" unicycle frames with disc brake mounts and a unique twin plate crown design.

Bedford
Bedford unicycles are a brand sold in Canada, specializing in trials unicycles; producing unicycles from beginner to pro.

Coker
The Coker Tire Company makes a 36" wheel unicycle, "The Big One", which is extremely popular, as for a long time it was the only unicycle available with this wheel size. Now the UDC 36" (Radial 36"), Nimbus 36" and Qu-ax 36" are becoming more popular in Europe, perhaps because Coker unicycles are very hard to get hold of outside the US. 'Coker' is still sometimes loosely used to refer to any 36"er.

Corbin
Corbin Dunn (USA) hand makes one-off custom unicycles, including the Corbin V-36 frame, tandem unicycle, and recumbent unicycle.

DM
David Mariner (UK) produced high quality giraffe, freestyle and mountain unicycles with frames from aircraft-grade steel. He also made custom unicycles. Unfortunately no longer in business, though the unicycles are still going strong.

Flansberrium
Jakob Flansberry (CA) handmakes his aluminium frames to order. Known for his expert welding, Flansberrium frames satisfy the irregular wheel market in unicycles. Fat tyre frames and 27.5inch wheels are a few of the frames to have come out of the Flansberrium workshop.

GB4
GB4 Manufacturing produces unicycle parts which are available exclusively through unicycle.com.

Hunter
Handcrafted by Hunter Cycles and often made to order. More expensive than mass produced unicycles, but highly praised by Muni riders. Frames are usually in the split "V shape."

Koxx
Koxx is a new brand in the unicycling scene. Koxx are known for very good bikes especially their trials bikes. Together with Yoggi, one of the best and most extreme unicyclists in France, they molded Koxx into a very fabulous new unicycle brand. They use some proven Try All parts like rims and tires.

Kris Holm
Kris Holm Unicycles, run by one of the world's leading trials and muni experts, produces several lines of trials and mountain unicycles which are very highly regarded. They recently announced a geared hub, co-produced with Schlumpf.

LiveWire
LiveWire Unicycles has developed a reputation over the past several years as a builder of the finest custom unicycles available. LiveWire scours the bicycling and unicycling world for cutting-edge parts and concepts and builds them into tough, beautiful unicycles, one customer at a time. LiveWire's specialties include 29" off-road racing unicycles, 36" touring unicycles, and 24" MUni wheels.

Mad4One
Mad4One, a premium unicycle company based in Italy, produces Street, Trials, Freestyle, MUni, and Race style unicycles. Very good quality, long lasting, high durability parts. Parts and full setups come with extensive custom color options via Mad4One's in house anodizing process.

Miyata
Miyata, a highly regarded and long established Japanese cycle manufacturer produces several different 20" freestyle unicycles, as well as a 24" model. Miyata unicycles are much more expensive than some other makers, but are very good quality.

Nimbus
Nimbus Unicycles produce a high quality range covering all styles of unicycling; from freestyle and commuting to Muni and Trials. The range also includes ultimate wheels, BC wheels and giraffes. They have recently developed their own ISIS hub for their Street, Freestyle and Trials unicycles as well as being one of the few manufacturers to have switched to frames with machined bearing cups across their range.

Onza
Onza are a trials bike manufacturer who have branched out into making really strong Trials and Muni (offroad) unicycles, and have recently produced a really strong ISIS splined hub to fit their famous Tensile cranks.

Pashley
A manufacturer of hand-made bikes in the UK, they made the first commercially-available muni (offroad unicycle), called the Pashley Muni, in the late 90s.

Pichlerrad
Pichlerrad are a German company which makes unicycles, ultimate wheels, giraffes and some bikes. Very good quality - these unicycles will last for years. Their 6 foot giraffe is a particularly good model.

Profile
Profile are a hub and crank manufacturer who produced one of the first splined hubs for unicycles, to try and cope with the abuse thrown at unicycles by Muni and trials unicycle riders. Their hub has inspired a loyal following with its limited lifetime guarantee for the cranks.

QU-AX
QU-AX are a German manufacturer. They produce unicycles of both splined and unsplined varieties. Their splined munis and trials unicycles are generally agreed to be about as strong as a KH or Onza. They are substantially cheaper.

Raleigh
A UK bicycle manufacturer who used to build their bikes in Nottingham but now gets them made in the Far East, has a basic unicycle in their range of bicycles but is more concerned with the bicycle end of the market. Raleigh is often sold under the Avenir brand name in the US.

Schlumpf
Revolutionary unicycle with geared hub, the first production geared unicycle and can change gear while riding.

Schwinn
Schwinn produced unicycles years ago, but stopped for financial reasons. Recently, however, they have reintroduced their unicycle line. Schwinn unicycles tend to be stronger than some other brands, but also heavier. Schwinn produces 20 and 24 inch models as well as the Giraffe (which is claimed to be the exclusive name of the Schwinn giraffe unicycles).

Semcycle
Designed by unicyclist Sem Abrahams, Semcycle unicycles come in several varieties, including freestyle, giraffe, multi-wheeled, and off-road models. The freestyle models are the most popular.

Siegmono Cycle
Siegmono produces in Germany and has been selling unicycles for 20 years. Besides standard sizes they have 12", 18", 22" and 30" unicycles; giraffes, multi-wheeled, excenter unis, and handlebar unicycles.

Sun
A low-cost brand of unicycles mostly used by beginners. Their unicycles have a reputation for being easy to break, particularly for larger wheel sizes.

Torker
The Torker Bicycle Company produces 5 different lines of unicycles; CX, LX, DX, TX and AX. The LX and CX are two of the most popular first unicycles in the US.

CX stands for classic, and is produced with 16", 20",24", and 30" wheel sizes. The rim is designed for basic riding and light freestyle; it will not put up with rough use.

LX stands for luxury, and even for beginners is a much better choice than the CX. It has a fairly comfortable saddle, a stronger 48-spoke wheel, and a wider frame. It is mostly used and most commonly known for freestyle, as well as very light trials and MUni.

DX stands for deluxe. This is Torker's heaviest-duty unicycle, suitable for extreme trials, MUni, and some street if desired.

TX means tall. Torker's five-foot giraffe is one of the least expensive available.

AX stands for Aluminium. Comes in 20", 24" and 29" wheel sizes. It has a lightweight aluminum alloy frame and has been discontinued.

Wyganowski
Hand crafted unicycles made in Minnesota, U.S.A. Building since 1995 building unicycles to meet the demands of professional riders. Each is designed and fit for the individual rider. Freestyle, Two Wheelers, 24" racing, Muni, 29er, and Giraffes up to 12 feet including pull-apart styles. Unicycle are built with top grade materials from the U.S. and Europe. Manufacturer of a heavy duty bearing holder. Paul worked in conjunction with T.C.U.C., Dustin Kelm, and Andy Cotter developing his freestyle, and touring unicycles during the origination of unicycle touring. He will soon be offering 100% Stainless steel unicycles built from Reynolds 953 and KVA stainless steel tubings as well as Titanium.

Yuni
A Unicycle.com house brand. The Yuni frame is narrower at the top than the Nimbus, which gets wider as it goes up and then expands into a baillorette

title: The Unicyclopedia/Manufacturers


text:
User:Mirv~enwikibooks

Please see my userpage on the English Wikipedia and my userpage on Meta.

title: User:Mirv~enwikibooks


text:
User talk:Mirv~enwikibooks

messages to w:en:User talk:Mirv please. I probably won't be checking in here very often. Mirv 07:16, 3 Feb 2005 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Mirv. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Mirv~enwikibooks that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:28, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 05:15, 19 April 2015 (UTC)

title: User talk:Mirv~enwikibooks


text:
User talk:Dcordero7965~enwikibooks

Hi, Dcordero,

It is nice that you update the development stages in the US history book, but take care of the following two things: 1) Change the date as well (update it to today when you change the stage), and 2) If there are red links (because the text was copied from Wikipedia without modification) either the links should be changed to link to Wikipedia (prefix w:) or remove the linking. Otherwise the chapter should not get 100% yet. (also the current links pointing to Wikibooks do not make any sense, because they are not in the US history book chapter structure). Thanks, --Andreas 08:44, 3 Feb 2005 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Dcordero7965. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Dcordero7965~enwikibooks that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:04, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 04:52, 19 April 2015 (UTC)

title: User talk:Dcordero7965~enwikibooks


text:
Movie Making Manual/Short Films/Fresh Films Cafe

Fresh Film's independent, film café 2005 programme is here. All events from Fresh Films London Ltd are not for profit. Look out for:
 * 1) Regular screenings on Tuesdays
 * 2) Children’s Film Festival and a Music and Film season both to be held in March and April 2005. This season is generously funded by Film London and New Cross Gate New Deal for the Community. The above programme is subsidised, supported by the National Lottery through the UK Film Council and Film London Regional Investment Fund for England. Full details to follow shortly.  We are now open for submissions for shorts and B movies

Tuesday screenings:

February 8th 2005:
and touching film. Paths not taken and lives not tasted are meshed in this part western, part odd couple film which manages to be funny and subtle whilst never straying into farce. As Hallyday cases the local bank his host prepares for surgery and the two develop an unspoken interconnection. A film to be savoured over wine.
 * L’homme Du Train: 2002. Patrice Leconte, starring Jean Rochefort and Johnny Hallyday.
 * ‘A charming melancholic divertissement’ Time Out. It’s a beautiful

 * February 22nd: La Haine (second in our online film poll)

 * March 8th:Delicatessen (winner of the online film poll)

£5.00 gets you a hot meal and entrance. The Café is fully licensed; bottled beer and wine available plus Fair Trade tea and espresso/ hot chocolate. Fabulous cake.

Films start at 8.00, café open from 7.00. Call Chris for more info: 07905 961 876. Or phone the café on: 020 8320 2317. Tickets can be bought in advance from the café. Early arrival advised, it gets packed.

More info: www.freshfilmslondon.com

=Where is it?= Café Crema, 306 New Cross Road. Close to New Cross and New Cross Gate tube stations on the East London Line. Easy on the bus.

=User's reviews=

/Short films /Sales and exhibition


text:
Movie Making Manual/Short Films/Exploding Cinema

More info: ExplodingCinema.org

=Reviews= I've only been to one Exploding Cinema event but it was excellent. Apparently they show EVERY film they're sent, which I find amazing because the films were some of the best I've seen for a long while. A thrilling mix of all sorts of films: from obscure and arty to comic and mainstream. They EC crew did a great job with the venue - they had loads of old 8mm projects setup all round the place playing loops of 70s porn and stuff! Brilliant night out - highly recommended. - Dan AKA Jack 03-02-2005

/Short films /Sales and exhibition


text:
Movie Making Manual/Short Films/ShortiCulture

Where: Ginglik, Shephard's Bush

More info: www.ginglik.co.uk

=Reviews= I love this shorts night. The films are always excellent quality and the venue is cosy and welcoming. My only gripe is that it can get VERY crowded - get there early! - Dan AKA Jack 03-02-2005

/Short films /Sales and exhibition


text:
User:Cauri

cauri likes projects. cauri created the Movie Making Manual and set up the basic structure and graphics.

title: User:Cauri


text:
High School Mathematics Extensions/Discrete Probability

Introduction
Probability theory is one of the most widely applicable mathematical theories. It deals with uncertainty and teaches you how to manage it.

Please do not misunderstand: We are not learning to predict things; rather, we learn to utilise predicted chances and make them useful. Therefore, we don't care about questions like what is the probability it will rain tomorrow?, but given that the probability is 60% we can make deductions, the easiest of which is the probability it will not rain tomorrow is 40%.

As suggested above, a probability is a percentage, and it's between 0% and 100% (inclusive). Mathematicians like to express a probability as a proportion, i.e. as a number between 0 and 1. So the probability that it will not rain tomorrow is 0.4.

Application
You might ask why we are even studying probability. Let's see a very quick example of probability in action.

Consider the following gambling game: Toss a coin; if it's heads, I give you $1; if it's tails, you give me $2. You will easily notice that it is not a fair game - the chances are the same (50%-50%) but the rewards are different. Even though we are playing with probability, there are useful, and sometimes not so obvious, conclusions we can make: one of them is that in the long run I will become richer and you will become poorer.

Another real-life example: I observed one day that there are dark clouds outside. So I asked myself, should I bring an umbrella? I use my observation of dark clouds as per my usual daily deciding routine. Since in past experiences, dark clouds are early warning signs of rain, I am more likely to bring an umbrella.

In real life, probability theory is heavily used in risk analysis by economists, businesses, insurance companies, governments, etc. An even wider usage is its application as the basis of statistics, which is the main basis of all scientific research. Two branches of physics have their bases tied in probability. One is clearly identified by its name: statistical mechanics. The other is quantum physics.

Why discrete probability?
 There are two kinds of probability: discrete and continuous. The continuous case is considered to be more difficult to understand, and much less intuitive, than discrete probability, and it requires knowledge of calculus. But we will touch on a little bit of the continuous case later on in the chapter.

Event and Probability
Roughly, an event is something we can assign a probability to. For example the probability it will rain tomorrow is 0.6; here, the event is it will rain tomorrow, and the assigned probability is 0.6. We can write
 * P(it will rain tomorrow) = 0.6

Mathematicians typically use abstract letters to represent events. In this case we choose A to represent the event it will rain tomorrow, so the above expression can be written as:
 * P(A) = 0.6

Another example is a (six-sided) fair dice will turn up 1, 2, 3, 4, 5 or 6 with equal probability each time it is tossed. Let B be the event that it turns up 1 in the next toss. We write:
 * P(B) = 1/6

Misconception  Please note that the probability 1/6 does not mean that it will turn up 1 in at most six tries. Its precise meaning will be discussed later on in the chapter. Roughly, it means that on the long run (i.e. the dice being tossed a large number of times), the proportion of 1s will be very close to 1/6.

Impossible and certain events
Two types of events are special. One type are the impossible events (e.g., a roll of a dice will turn up 7); the other type are certain to happen (e.g., a roll of a dice will turn up as one of 1, 2, 3, 4, 5 or 6). The probability of an impossible event is 0, while that of a certain event is 1. We write
 * P(Impossible event) = 0
 * P(Certain event) = 1

The above reinforces a very important principle concerning probability. Namely, the range of probability is between 0 and 1. You can never have a probability of 2.5! So remember the following
 * $$0 \leq P(E) \leq 1$$

for all events $$E$$.

Complement of an event
A most useful concept is the complement of an event. Here we use $$\overline{E}$$ to represent the event that the dice will NOT turn up 1 in the next toss. Generally, putting a bar over a variable (that represents an event) means the opposite of that event. In the above case of a dice:
 * $$P(\overline{E}) = 5/6$$

it means The probability that the dice will turn up 2, 3, 4, 5 or 6 in the next toss (notwithstanding the aforementioned misconception that a probability of X/Y does not mean that it will turn up X in at most Y tries) is 5/6. Please note that
 * $$P(\overline{E}) = 1 - P(E)$$

for any event E.

There are some other notations for (ways to write) complement rather than putting a bar (line) on top: prime (A') and star (A*). Both A' and A* mean: $$\overline{A}$$

Combining independent probabilities
Independent probabilities can be combined to yield probabilities for more complex events. I stress the word independent here, because the following demonstrations will not work without that requirement. The exact meaning of the word will be discussed a little later on in the chapter, and we will show why independence is important in Exercise 10 of this section.

Adding probabilities
Probabilities are added together whenever a single event can occur in multiple "ways". As this is a rather loose concept, the following example may be helpful. Consider rolling a single dice; if we want to calculate the probability for, say, rolling an odd number, we must add up the probabilities for all the "ways" in which this can happen -- rolling a 1, 3, or 5. Consequently, we come to the following calculation:
 * P(rolling an odd number) = P(rolling a 1) + P(rolling a 3) + P(rolling a 5) = 1/6 + 1/6 + 1/6 = 3/6 = 1/2 = 0.5

Note that the addition of probabilities is often associated with the use of the word "or" -- whenever we say that some event E consist of the events X, Y, or Z (being satisfied if any of the events occur) we use addition to combine their probabilities (if they are disjoint, see below).

A rule of thumb is that the probability of an event and the probability of its complement must add up to 1. This makes sense, since we intuitively believe that events, when well-defined, must either happen or not happen.

Multiplying probabilities
Probabilities are multiplied together whenever an event occurs in multiple "stages" or "steps." For example, consider rolling a single dice twice; the probability of rolling a 6 in two consecutive rolls (two times back to back) is calculated by multiplying the probabilities for the individual steps involved since the two events are independent. Intuitively, the first step is the first roll, and the second step is the second roll. Therefore, the final probability for rolling a 6 twice is as follows:
 * P(rolling a 6 twice) = P(rolling a 6 the first time)$$\times$$P(rolling a 6 the second time) = $$\frac{1}{6}\times\frac{1}{6}$$ = 1/36 $$\approx$$ 0.028 (or 2.8%)

Similarly, note that the multiplication of probabilities is often associated with the use of the word "and" -- whenever we say that some event E is equivalent to all of the events X, Y, and Z occurring, we use multiplication to combine their probabilities (if they are independent).

Also, it is important to recognize that the product of multiple probabilities must be less than or equal to each of the individual probabilities, since probabilities are restricted to the range 0 through 1. This agrees with our intuitive notion that relatively complex events are usually less likely to occur.

Combining addition and multiplication
It is often necessary to use both of these operations simultaneously. Once again, consider a dice being rolled twice in succession. In contrast with the previous case, we will now consider the event of rolling two numbers that add up to 3. In this case, there are clearly two steps involved, and therefore multiplication will be used, but there are also multiple ways in which the event under consideration can occur, meaning addition must be involved as well. The dice could turn up 1 on the first roll and 2 on the second roll, or 2 on the first and 1 on the second. This leads to the following calculation:
 * P(rolling a sum of 3) = P(1 on 1st roll)$$\times$$P(2 on 2nd roll) + P(2 on 1st roll)$$\times$$P(1 on 2nd roll) = $$\frac{1}{6}\times\frac{1}{6}$$ + $$\frac{1}{6}\times\frac{1}{6}$$ = 1/18 $$\approx$$ 0.056 (or 5.6%)

This is only a simple example, and the addition and multiplication of probabilities can be used to calculate much more complex probabilities.

Exercises
Let A represent the number that turns up in a (fair) dice roll, let C represent the number that turns up in a separate (fair) dice roll, and let B represent a card randomly picked out of a deck:

1. A dice is rolled. What is the probability of rolling a 3 i.e. calculate P(A = 3)?

2. A dice is rolled. What is the probability of rolling a 2, 3, or 5, i.e. calculate P(A = 2, 3 or 5)?

3. What is the probability of choosing a card of the suit Diamonds (in a 52-card deck)? . There are 4 suits, diamonds, spades, clubs, and hearts

4. A dice is rolled and a card is randomly picked from a deck of cards. What is the probability of rolling a 4 and picking the Ace of Spades, i.e. calculate P(A = 4)×P(B = Ace of spades).

5. Two dice are rolled together. What is the probability of getting a 1 and a 3?

6. Two dice are rolled separately. What is the probability of getting a 1 and a 3, regardless of order?

7. Calculate the probability of rolling two dice that add up to 7.

8. (Optional) Let C be the number rolled on the first dice and A be the number rolled on the second dice. Show that the probability of C being equal to A is 1/6.

9. Let C and A be as in exercise 8. What is the probability that C is greater than A?

10. Gareth was told that in his class 50% of the pupils play football, 30% play video games and 30% study mathematics. So if he was to choose a student from the class randomly, he calculated the probability that the student plays football, plays video games, and studies mathematics is 50% + 30% + 30% = 1/2 + 3/10 + 3/10 = 11/10. But all probabilities should be between 0 and 1. What mistake did Gareth make?

 Solutions

1. P(A = 3) = 1/6

2. P(A = 2) + P(A = 3) + P(A = 5) = 1/6 + 1/6 + 1/6 = 1/2

3. P(B = Ace of Diamonds) + ... + P(B = King of Diamonds) = 13 × 1/52 = 1/4

4. P(A = 4) × P(B = Ace of Spades) = 1/6 × 1/52 = 1/312

5. P(A = 1) × P(C = 3) + P(A = 3) × P(C = 1) = 1/36 + 1/36 = 1/18

6. P(A = 1) × P(C = 3) + P(A = 3) × P(C = 1) = 1/36 + 1/36 = 1/18 This is the same answer as the problem above because in both cases the outcome for each individual dice remains independent of the other regardless of whether or not they are thrown simultaneously. Another way of calculating the same answer is to consider that the first dice can be a one or a three but the second can only be one number - the opposite of the first dice, i.e. a 3 if the first dice was 1, or a 1 if the first dice was 3. That gives: P(A=1 or A=3) x P(opposite) = 2/6 x 1/6 = 2/36 = 1/18.

7. Here are the possible combinations: 1 + 6 = 2 + 5 = 3 + 4 = 7. Probability of getting each of the combinations are 1/18 as in exercise 6. There are 3 such combinations, so the probability is 3 × 1/18 = 1/6.

8. As C is the first dice rolled and can be any value, P(C) = 1. Given some value of the first role, P(A) = 1/6. The probability C and A have the same value is 1 * 1/6 = 1/6.

9. The probability of (C equal to A) is 1/6. Thus the probability of (C not equal to A) is 5/6. Half of such cases will be (C greater than A). Thus the probability of (C greater than A) is 5/12. 10. These three sets overlap so, for example, to get the probability of someone belonging to all three sets, you need to multiply (assuming they are independent), not add. P(F and V and M) = .5 x .3 x .3 = 0.045. It is necessary to remember that the events of playing football, playing video games, studying mathematics, or being human, a male, living in Armenia, etc are all possible. Although the likelihood and independence of these events/states may be debatable, the fact that the probability of any strange combination must be less than one must hold.

Random Variables
A random experiment, such as throwing a dice or tossing a coin, is a process that produces some uncertain outcome. We also require that a random experiment can be repeated easily. In this section we shall start using a capital letter to represent the outcome of a random experiment. For example, let D be the outcome of a dice roll. D could take the value 1, 2, 3, 4, 5 or 6, but it is uncertain. We say D is a discrete random variable. Suppose now that I throw a dice, and it turns up 5. We say the observed value of D is 5.

A random variable is the outcome of a certain random experiment. It is usually denoted by a CAPITAL letter, but its observed value is not. For example let
 * $$D_1, D_2, ..., D_n$$

denote the outcome of n dice throws, then we usually use
 * $$d_1, d_2, ..., d_n$$

to denote the observed values of each Di.

From here on, random variable may be abbreviated as "rv" (a common abbreviation in other probability texts).

The Bernoulli experiment
(This section is optional and it assumes knowledge of binomial expansion.)

A coin-toss is a simpler, specific form of the Bernoulli experiment. If we toss a coin, we will expect to get a head or a tail equally probably. A Bernoulli experiment is slightly more versatile than that, in that the two possible outcomes need not have the same probability.

In a Bernoulli experiment you will either get a
 * success, denoted by 1, with probability p (where p is a number between 0 and 1)

or a
 * failure, denoted by 0, with probability 1 - p.

If the random variable B is the outcome of a Bernoulli experiment, and the probability of a successful outcome of B is p, we say B comes from a Bernoulli distribution with success probability p (where $X \sim D$ means that the random variable X has the probability distribution D):

 * $$B \sim Ber(p) $$

For example, if
 * $$C \sim Ber(0.65) $$

then
 * P(C = 1) = 0.65

and
 * P(C = 0) = 1 - 0.65 = 0.35

Binomial Distribution
If we repeat a Bernoulli experiment n times and count the number of successes, we get a binomial distribution. For example:
 * $$C_i \sim Ber(p)$$

for i = 1, 2, ..., n. That is, there are n variables C1, C2, ... , Cn and they all come from the same Bernoulli distribution. We consider:
 * $$B =  C_1 + C_2 + ... + C_n$$

, then B is the random variable that counts the number of successes in n trials (experiments). Such a variable is called a binomial variable, and we write
 * $$B \sim Bin(n,p)$$

 Example 1

Aditya, Sarah, and John are equally able. Their probability of scoring 100 in an exam follows a Bernoulli distribution with success probability 0.9. What is the probability of
 * i) Only one of them getting 100?
 * ii) Two of them getting 100?
 * iii) All 3 getting 100?
 * iv) None getting 100?

 Solution

We are dealing with a binomial variable, which we will call B. And
 * $$B \sim Bin(3,0.9)$$

i) Aditya's (as well as Sarah and John's) probability of scoring 100 is 0.9 or 90%. We can write this as
 * $$P(S = 100) = 0.9$$

... where S represents the score of any of them. The probability of any of them getting 100 (success) and the other two getting below 100 (failure) is
 * $$0.9 \times 0.1 \times 0.1 = 0.009$$

but there are 3 possible candidates for getting 100, so
 * $$P(B = 1) = 3\times 0.009 = 0.027$$

ii) We want to calculate
 * $$P(B = 2)$$

The probability is
 * $$0.9 \times 0.9 \times 0.1 = 0.081$$

but there are $${3\choose 2} $$ combinations of candidates for getting 100, so
 * $$P(B = 2) = {3\choose 2} \times 0.081 = 0.243$$

iii) To calculate
 * $$P(B = 3) = 0.9 \times 0.9 \times 0.9 = 0.729$$

iv) The probability of "None getting 100" is getting 0 success, so
 * $$P(B = 0) = 0.1 \times 0.1 \times 0.1 = 0.001$$

The above example strongly hints at the fact the binomial distribution is connected with the binomial expansion. The following result regarding the binomial distribution is provided without proof; the reader is encouraged to check its correctness.

If
 * $$B \sim Bin(n,p)$$

then
 * $$P(B = k) = {n \choose k} p^k (1-p)^{n-k}$$

This is the kth term of the binomial expansion of (p + q)n, where q = 1 - p.

Events
In the previous sections, we have slightly abused the word "event". An event should be thought of as a collection (set) of outcomes of a certain random variable and hence we may assign a probability to it.

Let us introduce some notation first. Let A and B be two events, we define
 * $$\, A \cap B $$

to be the event of A and B. The probability of the event A and B is calculated as
 * $$\, P(A \cap B) = P(A) \times P(B) $$

We also define
 * $$ A \cup B $$

to be the event of A or B. As seen in exercise 10 above,
 * $$\, P(A \cup B) \ne P(A) + P(B) $$

in general. In fact,
 * $$\, 1 \geq P(A \cup B) \leq P(A) + P(B) $$

always holds.

Let's see some examples. Let A be the event of getting a number less than or equal to 4 when rolling a dice, and let B be the event of getting an odd number. Now
 * P(A) = 2/3

and
 * P(B) = 1/2

but the probability of A or B does not equal to the sum of the probabilities:
 * $$P(A \cup B) \ne P(A) + P(B) = \frac{1}{2} + \frac{2}{3} = \frac{7}{6}$$

as 7/6 is greater than 1.

It is not difficult to see that the event of throwing a 1 or 3 is included in both A and B. So if we simply add P(A) and P(B), some events' probabilities are being added twice.

The Venn diagram below should clarify the situation a little more,
 * [[image:HSE_Venn1.png|A or B]]

Think of the blue square as the probability of B and the yellow square as the probability of A. These two probabilities overlap, and the space where they overlap is the probability of A and B. So the probability of A or B should be:
 * $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

The above formula is the simplistic approach of the Inclusion-exclusion principle.

If for events A and B, we have
 * $$P(A \cap B) = 0$$

we say A and B are disjoint. This means both sets have no outcomes (elements) in common. If two events are disjoint, the following Venn diagram represents them:
 * [[image:HSE_Venn_2.png|A and B are disjoint]]

Venn diagram
 Traditionally, Venn diagrams are used to illustrate sets graphically. A set is simply a collection of things -- for instance, {1, 2, 3} is a set consisting of 1, 2 and 3. Venn diagrams are usually drawn round. It is generally very difficult to draw Venn diagrams for more than 3 intersecting sets. As an example, here is a Venn diagram showing four intersecting sets:

Expectation
The expectation of a random variable can be roughly thought of as the long-term average of the outcome of a certain repeatable random experiment, where by long-term average we mean that we perform the underlying experiment many times and average the outcomes. For example, let D be as above; the observed values of D (1,2 ... or 6) are equally likely to occur. So if you were to roll the dice a large number of times, you would expect each of the numbers to turn up roughly an equal number of times. So the expectation is
 * $$\frac{1 + 2 + 3 + 4 + 5 + 6}{6} = 3.5$$

We denote the expectation of D by E(D), so
 * $$ E(D) = 3.5 $$

We should now properly define the expectation.

Consider a random variable R, and suppose the possible values it can take are r1, r2, r3, ..., rn. We define the expectation to be
 * $$E(R) = r_1P(R = r_1) + r_2P(R = r_2) + ... + r_nP(R = r_n)$$

Think about it: Taking into account that the expectation is the long term average of the outcomes, can you explain why E(R) is defined the way it is?

 Example 1  In a fair coin toss, let 1 represent tossing a head and 0 a tail. The same coin is tossed 8 times. Let C be a random variable representing the number of heads in 8 tosses. What is the expectation of C, i.e. calculate E(C)?

Ans. E(C)=∑[r x P(C=r)]  where    0<=r<=8

$$ \begin{align} P(r) &= \binom{8}{r} \cdot \left ( \frac{1}{2} \right )^r \cdot \left ( 1 - \frac{1}{2} \right )^{8 - r} \\ &= \binom{8}{r} \cdot \left( \frac{1}{2} \right )^8 \\ E(C) &= 0 \cdot \binom{8}{0} \cdot \left ( \frac{1}{2} \right )^8 + 1 \cdot \binom{8}{1} \cdot \left ( \frac{1}{2} \right )^8 + \dots + 8 \cdot \binom{8}{8} \cdot \left ( \frac{1}{2} \right )^8 \\ &= (0 + 8 + 56 + 168 + 280 + 280 + 168 + 56 + 8) \cdot \left ( \frac{1}{2} \right )^8 \\ &= 1024 \cdot \frac{1}{256} \\ &= 4 \\ \end{align} $$

So the expectation value is 4

Areas as probability
The uniform distributions...

Order Statistics
Estimate the x in U[0, x]. ...

Addition of the Uniform distribution
Adding U[0,1]'s and introduce the CLT.

...CLT - Central Limit Theorem: In any set of sample distributions, as the number of samples taken increases, the overall mean distribution of the sample distributions will approach a Normal distribution.

The CLT is important in Statistical inference where small samples are taken of entire populations to draw conclusions on the entire population.


text:
Talk:High School Mathematics Extensions/Discrete Probability

Ok, to tell you the truth: I do not know much about discrete probabilities. I'm very interested to see how it will develop. I've just seen a A.Level maths exam paper, and the first question is about probability.

P(A)=0.8,P(B)=0.75,P(A"intersection"B)=k.

a)Express P(A"Union"B) in terms of k.

(My)Answer:We sum up P(A) and P(B), but then the probabilities that both A and B are true are counted twice, therefore we subtract k. So it is 0.8+0.75-k=1.55-k

b)Hence show that 0.55≤k≤1

(My)Answer:Since all probabilities are smaller than or equal to 1, k must be at least 0.55. Since k itself is also a probability, it must be smaller than or equal to 1. Putting these together, we get 0.55≤k≤1

...

--Lemontea 11:49, 3 Feb 2005 (UTC)

 * All correct, and explained very well. To bring your maths to uni level (which you are more than capable of) you need to quote the De Morgan's laws for part a) and actually solve 0 <= 1.55 - k <= 1 algebraicly and use the fact that 0 <= k <=1, for part b). 10/10. Xiaodai 00:25, 6 Feb 2005 (UTC)

I've added some information on adding and multiplying probabilities -- I hope it's all okay. Comments and revisions, of course, are welcome. On another note, I wanted to ask if anyone knows what the final section on options pricing has to do with discrete probability. I personally can't find anything relevant in it.
 * --Buggi22 07:08, 26 May 2005 (UTC)

 * That's okay, and it's an essential part of the topic. However, may I point out that the multiplacation law of probability does not always apply: It is true only when they are independent events. But then, how will we introduce the concept of dependent and independent events, conditional probability etc? Should we put them before or after your section. I'm at a lost at this moment and are looking forward for your work. --Lemontea 09:42, 26 May 2005 (UTC)

 * Option pricing is an important part of finance mathematics. Although it doesn't really use probability in the true sense, it does contain some material on "arbitrage free" probabilities which i think is interesting. Xiaodai 11:08, 27 May 2005 (UTC)

Tree diagram
why there are no three diagrams? 06:14, 22 October 2009 (UTC)

Misconception
I believe there is bias here. I am not expert on the field so I don't dare to edit the page directly. I am talking about this part:

Please note that the probability 1/6 does not mean that it will turn up 1 in at most six tries. Its precise meaning will be discussed later on in the chapter. Roughly, it just means that on the long run (i.e. the die being tossed a large number of times), the proportion of 1's will be very close to 1/6.

There are two school of thought that interpret what does probability means:

http://en.wikipedia.org/wiki/Probability#Interpretations

That misconception just reflect the 1st school of thought, the frequentist while Bayesian school of thought is not properly represented here. 05:56, 22 October 2009 (UTC)05:56, 22 October 2009 (UTC)ArielGenesis (talk) 05:56, 22 October 2009 (UTC)

probability space section feels clunky
I have pasted the following thing here so we can re-use the material later if need be:

We have now all the machinery required to define a Probability Space. Before we do that, let's consider the example of throwing a die again. If we throw a die, the numbers that will turn up are 1, 2, 3, 4, 5 or 6, and nothing else. Notice the events are disjoint and their probabilities add up to 1, i.e. all the possible events have been accounted for.

Definition -- Probability Space
A finite probability space of a random experiment consists of events:
 * $$E_1, E_2,....E_n$$

such that:
 * 1) $$P(E_1) + P(E_2) + ... + P(E_n) = 1$$
 * 2) all events are disjoint

In the above definition a random experiment is simply an action like tossing a coin or picking a card, where the outcome is uncertain. For example, let the random experiment be tossing a die, and let the events be the die turning up 1, 2, 3, 4, 5, or 6. So there are 6 events, and they are all disjoint, and the probabilities of each of the events add up to 1, i.e. all the possible events are included. Therefore it's a probability space.

Example 1

Let the random experiment be choosing a card randomly from a complete deck. And let the events be the card is of the suit Spades, Hearts, Clubs or Diamonds. Is this a probability space?

Solution To check whether it is a probability space, we need to check two conditions. so it's a probability space.
 * 1) There are 4 events, and their probabilities add up to 1.
 * 2) Each of the events are disjoint

Example 2

Let the random experiment be tossing a coin twice. Let the events be
 * the first toss results in a head, or
 * the second toss results in a tail

Is this a probability space?

Solution We proceed to check the two conditions:
 * 1) There are 2 events. Each event has probability of 1/2 (check) and so they add up to 1.
 * 2) The events are not disjoint! So it is NOT a probability space.

Exercises
1. Let the random experiment be tossing a coin once. Let the 2 events be tossing H or T. Is this a probability space

2. Let the random experiment be tossing a die three time. Let the 3 events be the maximum of the 3 die throws are . Is this a probability space
 * 1) 1 or 2,
 * 2) 3 or 4, or
 * 3) 5 or 6

3. Give an exmaple of an experiment that satisfies the first condition but not the second.

4. Give an exmaple of an experiment that satisfies the second condition but not the first.

5. Let S be a probability space. It has n events
 * $$ E_1, E_2, ... E_n$$

Use the Simple Inclusion Exclusion Principle, show that
 * $$ P(E_i \cup E_j) = P(E_i) + P(E_j)$$

for i ≠ j.

6. Is it true that the complement of any event Ei is the union of all the other events i.e. is the following true:
 * $$ \overline{E_i} = E_1 \cup E_2 \cup ... \cup E_i \cup E_{i+1} \cup ... \cup E_n$$

(Hint: use $$ P(\overline{E_1}) = 1 - P(E_1)$$)--Xiaodai 05:57, 6 August 2005 (UTC)

how to pronounce...
Could someone please add some verbiage (in parentheses in the article) that explains how to say/pronounce/express-in-words things like X ~ B.   Is that pronounced "x has a distribution of b"? What about B ~ Bin(n,p)? This would be helpful when trying to converse in person with other mathematicians. Thanks

Impossible and Certain Events equation seems off
Since an answer can only be 0-1, then 0 <= P(E) <= 1 seems wrong, since it would be 0 =>. Greater Than Or Equal To 0, rather than Less Than Or Equal To. As it is, the equation simply says that the event is either 1 or less than 1. Looks like it anyway, am I wrong or is the image?

one addition to help completeness
It's been a while since I studied statistics. I found your presentation easy to follow and purposeful. You used the term (3 over 2) and (n over k). I can't recall what the calculation is for that terminology. And since it is referenced, I don't have an easy way to look it up. Adding a line defining that function would be useful.

Review answer 5
I guess that the question 5 has a mistake on its answer.

As the order doen't matter the P(A=1 and B=3) + P(A=3 and B=1) = 1/6 x 1/6 + 1/6 x 1/6 = 1/18 - Is this now question 6?

''6. Two dice are rolled seperately (sic). What is the probability of getting a 1 and a 3, regardless of order?''

I agree that this seems incorrect, or at least the explanation of the result is inaccurate: the first roll can be either a 1 or a 3. The answer is not as simple as repeating the first half twice: once a result is recorded for the first roll, that result can not be repeated in the second roll; otherwise the result could be two 1s or two 3s. Instead, the probability for the second roll must be 1/6, as it must be either a 1 or a 3, but only whichever of those two was not rolled first. So, if I'm not mistaken, the final probability must be: (1/6 + 1/6) x (1/6) = 2/36 or 1/18

Review answer 10
It looks like answer 10 has mistake. Gareth calculates probability of choosing the student that either plays football or plays video games or studies mathematics. This probability cannot be 11/10 but it cannot be .045 either.

--Deesto (discuss • contribs) 14:54, 31 July 2012 (UTC)

Review Answer 7
I'm a little confused about example number 7:

''Here are the possible combinations: 1 + 6 = 2 + 5 = 3 + 4 = 7"

I think that assumption is correct, but the next one:

"Probability of getting each of the combinations are 1/18 as in exercise 6''

I think it's not:

$$P(A=1) x P(A=6) + P(A=2) x P(A=5) + P(A=3) x P(A=4) = 1/36 + 1/36 + 1/36 = 1/12$$

And the final resolution:

"There are 3 such combinations, so the probability is 3 × 1/18 = 1/6."

Should be:

$$3 x 1/12 = 3 / 12 = 1/4$$

Am I correct?

Thank you all!

 * No. Review the section "Combining addition and multiplication".
 * Instead of

 * $$P(A=1) \times P(A=6)$$ (and similarly for other terms)

 * use

 * $$(P(A=1) \times P(A=6) + P(A=6) \times P(A=1))$$ (and similarly for other terms)

 * so we get

 * $$(1/36 + 1/36) + (1/36 + 1/36) + (1/36 + 1/36) = 2/36 + 2/36 + 2/36 = 3 \times 1/18 = 1/6$$

 * or, alternatively (review Solution 6)

 * $$(P(A=1) + P(A=6)) \times P(opposite)$$

 * so we get

 * $$(2/6 \times 1/6) + (2/6 \times 1/6) + (2/6 \times 1/6) = 2/36 + 2/36 + 2/36 = 3 \times 1/18 = 1/6$$

 * Cartland (discuss • contribs) 05:58, 8 October 2012 (UTC)

Combining addition and multiplication
There is often confusion about the order of multiplying and adding and the resultant meaning. This adds some additional clarification to this section of the main text.

From the main text, we have
 * P(rolling a sum of 3) = P(1 on 1st roll)$$\times$$P(2 on 2nd roll) + P(2 on 1st roll)$$\times$$P(1 on 2nd roll) = $$\frac{1}{6}\times\frac{1}{6}$$ + $$\frac{1}{6}\times\frac{1}{6}$$ = 1/18 $$\approx$$ 0.056 (or 5.6%)

In what way is this different from
 * P(?) = (P(1 on 1st roll) + P(2 on 2nd roll)) $$\times$$ (P(2 on 1st roll) + P(1 on 2nd roll)) = ($$\frac{1}{6} + \frac{1}{6}$$) $$\times$$ ($$\frac{1}{6} + \frac{1}{6}$$) = $$\frac{2}{6}\times\frac{2}{6}$$ = 1/9 $$\approx$$ 0.111 (or 11.1%) ?

The short answer is because it's answering a different question: not P(rolling a sum of 3) but P(only 1 or 2 in both rolls).

Sample spaces can be useful to see what's going on (https://en.wikipedia.org/wiki/Probability_space).

For simplicity, consider a 4 sided dice (pyramid). The sample space of two rolls (or 2 dice rolled together), represented by single digit decimals, is (16 possibilities):

(Hopefully it is obvious that, for example, 23 is not twenty three, but represents 2 on one roll and 3 on the other roll ;).

P(rolling a sum of 3) contains 12 and 21

= P(1 on 1st roll)$$\times$$P(2 on 2nd roll) + P(2 on 1st roll)$$\times$$P(1 on 2nd roll) = $$\frac{1}{4}\times\frac{1}{4}$$ + $$\frac{1}{4}\times\frac{1}{4}$$ = $$\frac{1}{16} + \frac{1}{16}$$ = 2/16 = 1/8

P(only 1 or 2 in both rolls) contains 11 and 12 and 21 and 22

= (P(1 on 1st roll) + P(2 on 2nd roll)) $$\times$$ (P(2 on 1st roll) + P(1 on 2nd roll)) = ($$\frac{1}{4} + \frac{1}{4}$$) $$\times$$ ($$\frac{1}{4} + \frac{1}{4}$$) = $$\frac{2}{4}\times\frac{2}{4}$$ = 4/16 = 1/4

--Cartland (discuss • contribs) 07:02, 8 October 2012 (UTC)

No Coverage of Conditional Probability
I think this page is a useful starting point. Some points for improvement (and apologies if I have mistakes of notation or process - I'm new to this form of communication):

 * There are vague references at the end to distinctly non-discrete topics like the uniform distribution. Placeholders I could understand, but not to areas that are off topic for the this page.
 * Some coverage of conditional probablility would be useful. There are some simple concepts and notations that can be explained with the kind of venn diagram samples that are nicely used in parts of the existing material, e.g. Bayes Theorem.
 * I echo someone else's comment that (3 over 2) is a notation that needs to be introduced if it is going to be used. Better perhaps not to use it at that point in the text.

Please explain the notation
In the section The Bernoulli experiment we have the notation:

 * $$B \sim Ber(p) $$

what does the function $$ Ber(p) $$ mean? I am not sure what this section is saying, but I guess it is the probability of result p? -- Q Chris (discuss • contribs) 13:52, 8 November 2012 (UTC)

Venn diagram claim
I am confused by this claim in the Venn diagram section: "It is generally very difficult to draw Venn diagrams for more than 3 intersecting sets. To demonstrate why, here is a Venn diagram showing four intersecting sets." Essentially, this says "here is a diagram of something very difficult to draw", and yet there it is -- a fairly simple diagram. Am I misinterpreting this? --Ds13 (discuss • contribs) 23:10, 18 November 2012 (UTC)

It's rather hard to find intersection of all 4 sets. Try yourself. --ViolentOr (discuss • contribs) 13:42, 20 January 2013 (UTC)

Exercise 9
I think all is well explained and there is no too much content, but perhaps the proposed solution for the exercise 9 is a little tricky for some students. A more consistent solution with the exposed explanations could be:

P(C>A) = P([(C=1)⋂(A>1)] ⋃ [(C=2)⋂(A>2)] ⋃ [(C=3)⋂(A>3)] ⋃ [(C=4)⋂(A>4)] ⋃ [(C=5)⋂(A>5)]) ; Note: P(A>6) = 0;

P(C>A) = P(C=1).P(A>1)+P(C=2).P(A>2)+P(C=3).P(A>3)+P(C=4).P(A>4)+P(C=5).P(A>5);

Since P(C=1) = P(C=2) = P(C=3) = P(C=4) = P(C=5) = 1/6 = P(C=any) for 1 ≤ any ≤ 6 :

P(C>A) = P(C=any).(P(A>1)+P(A>2)+P(A>3)+P(A>4)+P(A>5));

And since P(A>i) = (6-i)/6 for 1 ≤ i ≤ 6 :

P(C>A) = 1/6.(5/6 + 4/6 + 3/6 + 2/6 + 1/6) = 15/36 = 5/12

A discussion of the Monte Hall "paradox" (from the gameshow) is always a good practical example.
http://en.wikipedia.org/wiki/Monty_Hall_problem

Optional exercise - ex. 7
Is this the only optional one? If it is the only optional exercise, wouldn't it make sense to move it to the end of the exercises. Otherwise it seems weird to have one exercise that is unecessary then more to do.

If ex.8 - ex.10 are also optional, they need marking as such.


text:
Talk:Portuguese

Welcome sentence
Why put the welcome sentence as this? I believe it is hard to read and to understand. I propose a change to something more like this: --Jotomicron 15:59, 3 Feb 2005 (UTC)

 * I agree. Especially since sentences don't really translate word-for-word like that most of the time (in real life). -- anon 12 April 2005

 * I've changed it so that all Portuguese text is together as suggested for readability -- anon 17 April 2005

Format
I suggest format this book like the books German and French. It will make this book more comprehensible and easy to maintain it. --User:Marlosfabris 02:01, 6 Oct 2006 (UTC)

 * There's a couple of other discussions over at Talk:Portuguese/Contents. Someone suggesting collaboration with Portugiesisch and other suggesting following the French and Spanish structure. Perhaps WikiProject Languages should lay down some guidelines for which is the best structure to follow -- Harry Wood (talk) 14:00, 13 January 2011 (UTC)

About Brazilian Portuguese on this Wikibook
Hello! I'm a newcomer to the wikibooks and a native Portuguese.

When I looked at the Portuguese Language Wikibook I understood it was written in Brazilian Portuguese.

I was wondering if Portuguese from Portugal could be used as it's the one I truly master and the only in which I could help the wikibook.

-- User:Portuguese15 31 March 2007

 * See discussion over at Talk:Portuguese/Contents -- Harry Wood (talk) 14:00, 13 January 2011 (UTC)

Galicia
Why is Galicia listed as a region where Portuguese is official? Galician and Portuguese being the same language is not the Xunta's official interpretation. Ungoliant MMDCCLXIV (discuss • contribs) 20:06, 2 December 2012 (UTC)
 * Any objections against me removing Galicia from the list? Ungoliant MMDCCLXIV (discuss • contribs) 17:17, 3 February 2013 (UTC)

title: Talk:Portuguese


text:
Movie Making Manual/Pitching/Synopsis

A synopsis is the kind of brief summary you might read on the back of a video box, and generally doesn’t run for more than one page. Without going into any great detail, it describes the main character and what they are up against, as well as the major turning points in the screenplay. A synopsis is a useful pitching tool, but it’s also a very useful way for screenwriters themselves to articulate what the story is basically about.

/Writing movies

title: Movie Making Manual/Pitching/Synopsis


text:
Movie Making Manual/Pitching/Logline

A logline is a very short, catchy summary of the story, usually not longer than one sentence. It serves as a way to communicate with great efficiency what a story or script is about. Typically a logline is used to help industry people quickly determine whether the story in question is one which a producer, director, or studio wishes to make. The logline first came into being in the script vaults of the old studios. Loglines were often written on the cover or even the spine of a screenplay, allowing a producer or director of a certain genre to quickly scan hundreds of scripts and narrow their search to just scripts which were in the genre they wished to produce. A good logline contains the following three parts: Writing a good logline is very difficult for most people because of a failure to understand the three basic tenets of a good logline as outlined here.
 * Who the Main Character is (MC, also known as the Protagonist).
 * What the Main Conflict is (i.e. what the MC wants and who or what is stopping them).
 * What is at Risk, if the MC fails.

/Writing movies

title: Movie Making Manual/Pitching/Logline


text:
Business English/Slang

No content here yet. Please look at this slang and idioms link while we get started on this page.

empty

A Ambulance chaser:			An unskilled and unethical lawyer. B Ball park:	A baseball term: To estimate how close something is to an accurate number. If something is in the ball park it is close enough to be acceptable. Basement Betty:			A woman that runs a business from her home. Belt tightening:				To reduce expenses. To save money. Bite the bullet:				To make a difficult decision. (a) Bitter pill to swallow:	To receive bad news. To hear something you don’t want to hear. Bottom line:	The truth behind what is being said. Being honest even if other people do not want to hear it. Brownie points:		Receiving credit for a good deed or giving a compliment. C Cash cow:	A product or service that creates a lot of money without much investment. (to) cash in:	To make money from or to benefit from financial or social investment. (to) Climb the corporate ladder:	To get promoted. To move upwards in the organization. Crunch time	A short period of time when there is a lot of work to do and a lot of responsibility. D Didn’t go over well:	When a plan or activity did not happen as planned and was not successful. (a) Dog eat dog world:	A mean and tough world where people only help themselves. Dot your i’s and cross your t’s:	Check your work. Pay attention to the details. Down to the wire:	Working to the last minute. To finish just before the deadline. (to) Drum up business:	To find new customers. (to) Face the music:	To admit there is a problem. To prepare for disciplinary action. E Egg on the face:	When something embarrassing happens to a person. When a person looks foolish. Elevator pitch:	A very short speech that explains yourself or your product in less than ten seconds. F Fast track:				To speed up a project. To give something a higher priority. Fly by the seat of your pants:	To do something without any planning and just react to what is happening. G Glitch:	A mistake or when something is not working correctly. This usually refers to a piece of technology. Golden parachute:	When a high member of management is fired and is given a very generous severance package. Gut feeling	An instinct or intuition that makes you expect how something will happen. H Hard sell:                                       	 Pushing hard to sell a product or service that is usually difficult to sell. (to) Have a lot on your plate:	To have a lot of work to do. To have a lot of responsibilities. Heads will roll:	An expression that means people will be disciplined harshly for a mistake. This may mean people will be fired. I I hear you:				I understand. J (to) Jump the gun:			To start something too soon or to begin before everyone else. K (to) Keep your eye on the prize:		To stay focused on the final results of a plan or project. (to) Keep it under wraps:		To keep a secret. To not tell other people about something. (to) Kick the bucket:	When something like a machine or a project is finished. It is used to suggest that something has died. L Lateralled:	A lateral transfer. To move to a new position without moving up or down the organization’s ranking. People often take lateral transfers to learn new skills. Lemon:	A product, usually a car, which has a problem and does not work properly. M N O Off the top of my head:			To think up ideas without and planning time. On the tip of my tongue:		When you know a word but can’t remember how to say it Out in left field:	A baseball term. To not understand what is happening. To not pay attention. Out of the loop:	To not understand what is happening because a person is not involved in the proper communications. P Page:					To call someone over the phone system in the office. (to) pass the buck:			To blame someone else for a problem or mistake. Playing ball:	A baseball term: Cooperating with clients and associates. Working together to complete a task. Playing hardball:		A baseball term. To get mean and tough. (to) Plug (a product):		To talk positively about a product socially (to) Pull the plug:		To stop or kill a project. (to) Pull your weight:		To do your share of the work. (at a) Premium:				To buy or sell something at the best price. (to) Push the envelope:		To move faster, work harder and do better than expected. Q R (to) Rally the troops:	To motivate the people around you. To energize people for a better performance. Reality check:	To think about how something will really work or if it will not work at all. Right off the bat:			A baseball term. To start something immediately. Right on target:				When a plan or activity happens perfectly. S (to) Scale back:	To reduce the number of something. This could be the number of hours worked, the number of employees, the amount of money paid in salaries, etc. (to)Shake on it:				Making a promise with a handshake. Show him/her the door:			To get rid of someone. To fire a person. Slam Dunk:				A basketball term: When something is perfect. (the) Squeaky wheel gets the grease:	Bringing attention to a problem so people will work to fix it. Stay on your toes:			Be careful. Be ready for trouble. (to) Step up to the plate:	A baseball term. This term is now often shortened so people will say ‘to step up.’ It means to take on a challenge, to do your best or to volunteer. T Temp:	A temporary worker. Someone that is hired by a staffing agency for a short time. Through the roof:	Numbers are higher than expected. U Use some elbow grease: 		To work hard to get something done. V

W (to) wear many hats:	To have different jobs and responsibilities in one or more organization. (to) Wing it:	To explain something without any planning. This could be giving a speech or explaining an idea. (the) Whole nine yards:	A football term: To ‘go the whole distance.’  To see a project through from start to end. (to) work out the kinks:	To solve the problems. To make sure something works properly. X Y Yes man:				An employee that always agrees with the boss. Z

title: Business English/Slang


text: Template:EM

title: Template:EM


text:
Template:EMDisclaimers

Please Read: Warnings and Disclaimers
/Templates
 * Please read the Wikibooks General Disclaimer and Medical Disclaimer.
 * Never post any material which contains Private Health Information (PHI). This is not only wrong, it will probably expose the person posting PHI to criminal sanctions and other professional repurcussions. Please alert a system administrator if you see anything you suspect to be PHI. If in doubt, report it or remove it.
 * Plagiarism, breach of copyright, impersonation, and vandalism are not OK.
 * Do not use this textbook as a guide to treating yourself or anyone else! Don't use it as a guide to any health decisions. Consult your doctor, don't make decisions using this book. The information in this textbook can not be trusted for making medical decisions.
 * The information in this textbook can be edited by anyone, regardless of whether they are medically trained or not, regardless of whether they are providing good information or misleading information. The information in this textbook may be wrong! Using this information to make health decisions for yourself or anyone in your care may lead to death, disability, pain and suffering, or worse!
 * The material in this textbook has not been checked for accuracy or safety
 * This wikibook is intended to be a collaborative communication between physicians, nurses, physician assistants, paramedics, EMTs and other healthcare providers. It is not intended for the general public. It is under development and inevitably contains many errors and omissions.

title: Template:EMDisclaimers


text:
Cookbook:Fluid Ounce

A fluid ounce is a unit of volume measure equal to 2 tablespoons, 6 teaspoons, or 1/8 cup. It is approximately equal to 30 ml.

The fluid ounce may be abbreviated as oz or, more specifically, as fl oz.

Note that the fluid ounce is distinct from the ounce.

title: Cookbook:Fluid Ounce


text:
Template:EMContribs

For Contributors
/Templates
 * Use one of these stencils as a model structure, whenever possible, to provide some uniformity in this textbook:
 * Standard Core Chapter Stencil
 * Reference Summary - Page Stencil - for One PubMed Paper
 * The ABEM/ACEP Model of the Clinical Practice of Emergency Medicine guides creation of chapters and topics in this textbook, wherever possible.
 * If any references are medical articles with Unique Identifiers, they may link out to the PubMed Abstract.
 * For instance, a link to the PubMed article with Unique Identifier = 15635313 is created a PubMed Link - simply use the prefix 'PMID' and the PubMedId Number, for example:
 * Respiratory Rate Measurement PMID 15635313
 * The PubMed home page is here:
 * National Library of Medicine - PubMed
 * If you are contributing for the first time, or if you are not familiar with the Wiki system, please read this:
 * Guide to the OCTEM
 * Miscellaneous pages:
 * Roles
 * PIOPED References

title: Template:EMContribs


text:
Cookbook:Pint

A US pint (that is, the fluid pint) is a unit of volume measure equal to 2 cups, ½ quart, or 16 fluid ounces. It is approximately equal to 473 ml. When converting a recipe to metric, it is usually sound to round the pint to 500 mL, along a pound of 500 g.

The dry pint, approximately 551 ml, is not commonly used in cooking. It is used in the sale of some items such as blueberries in the United States.

The Imperial pint is almost exactly 20% larger, 568 ml, and divided into 20 imperial fluid ounces rather than 16 US fluid ounces. (Note that US fluid ounces are slightly larger than imperial fluid ounces.)

title: Cookbook:Pint


text:
Cookbook:Quart

A quart (that is, the liquid quart) is a unit of volume measure equal to ¼ gallon, 4 cups, 2 pints, or 32 fluid ounces. A quart is approximately equal to 946 ml, or more roughly to 960 ml or 1 L.

The dry quart is somewhat larger, at about 1.1 L. It is not generally used in cooking.

The now unused British quart is also somewhat larger, containing 40 ounces of a slightly different size. The British quart is about 3% larger than the dry quart and 20% larger than a U.S. liquid quart.

title: Cookbook:Quart


text:
Cookbook:Gallon

A gallon (the Queen Anne gallon, as used in the USA) is a unit of volume measure equal to 4 quarts. A gallon is approximately equal to 3.8 L.

The post-1824 British gallon is about 20% larger. The Queen Anne (USA) gallon is thus 16.7% smaller, or 5/6 the size of a British gallon.

title: Cookbook:Gallon


text:
User:4.62.127.83

Any questions regarding this citation please forward them to RFORD at Rexfordshouse@yahoo.com

title: User:4.62.127.83


text:
Cookbook:Liter

The liter (AE) or litre (BE), abbreviated l (or sometimes 'L' in the US, Canada and Australia), is a metric unit of volume. It is equal to 1000 ml, also known as 1000 cm³ (informally cc). The litre is a bit larger than a US liquid quart. One liter of pure water weighs 1 kilogram at 4°Celsius.

As many recipes have been roughly converted from one system to another, a similarly approximate conversion back may yield the original recipe. When converting recipes from one system to another, quantities should always be rounded to the nearest reasonable unit to avoid awkward measurements, e.g. 11½ ounces and 347 grams should be rounded off to 12 ounces and 350 grams respectively, and so on.

title: Cookbook:Liter


text:
Cookbook:Milliliter

The milliliter (AE) or millilitre (BE), abbreviated mL or ml, is a metric unit of volume measurement. It is equal to the cubic centimeter (cm³ or colloqually cc) and to 0.001 L.

As many recipes have been crudely converted from one system to another, a similarly crude conversion back may yield the original recipe. Crude conversions are as follows:

title: Cookbook:Milliliter


text:
Cookbook:Gram

The gram or gramme (old BE), abbreviated g, is a unit of weight. It is equal to one thousandth ( ¹⁄ ₁,₀₀₀ [0.001]) of a kg. There are 454 ⁵⁄ ₁₁ (454. 54 ) g per pound. Since there are sixteen (16) ounces in a pound, there are 28 ⁹⁄ ₂₂ (28.4 09 ) g in one ounce. As many recipes have been crudely converted from one system to another, a similarly crude conversion back may yield the original recipe. (Cooking is not rocket science.)

title: Cookbook:Gram


text:
Cookbook:Pound

The pound, abbreviated lb (or #, exclusively in the US), is a unit of weight. It is divided into 16 ounces. One pound is equivalent to approximately 10/22 kg or 454 g in the metric system. When converting a recipe to metric, it is usually sound to round the pound to 450 g and the ounce to 30 g. A metric pound is equal to half a kilogram or 500 g and is sometimes seen in recipes originating on the continent.

title: Cookbook:Pound


text:
Cookbook:Ounce

The ounce (avoirdupois ) is a unit of mass ("weight") equal to $1/undefined$ of a pound or about 28 g. The ounce is herein abbreviated as oz. When converting a recipe to metric, it is usually sound to make the following approximations.

title: Cookbook:Ounce


text:
Portuguese/Contents/Number and Gender

Introduction
The Portuguese language has two grammatical categories shared by a large number of other languages, especially the Indo-European ones. These two categories are Number and Gender. As a speaker of English, you certainly know that the Number of a word expresses the amount of this word. Thus we have nouns like door and doors.

Speakers of Romance languages, German, Greek and a lot of others also understand what Gender is. In Portuguese, each noun has one specific gender, and unless this noun is related to people or animals, it is highly random. In English there are some examples of nouns with gender, like man and woman, actor and actress.

Unlike some languages, there is no case system in Portuguese, not even some vestiges of the case systems in Latin or other ancestor of the Language.

Number
There are two Numbers:
 * Singular - expresses the idea of a single object;
 * Plural - expresses the idea of several objects.

Gender
There are also two Genders: Nouns that are not related to any of those genders are distributed randomly across the two possibilities. Thus, the noun "casa", house is feminine, while "pão", bread is maculine.
 * Masculine - used on nouns related to men or male animals;
 * Feminine - used on nouns related to woman or female animals.

Tips for Identifying Word Gender
As said above, gender is mainly random. However, there are some ways to find out if a certain word is masculine or feminine, by looking at its ending:
 * Words ending with -o, -im, -om, -um, -or are usually masculine:
 * "carro" car;
 * "rim" kidney;
 * "som" sound;
 * "atum" tuna fish
 * "cantor" male singer.
 * Words ending with -a, -ice, -ção, -são, -gem, -ez, -iz are usually feminine
 * "casa" house;
 * "patetice" foolishness;
 * "Conceição" (Proper Name);
 * "coragem" courage;
 * "acidez" acidity
 * "actriz" actress.

Bear in mind that these rules are not flawless. For instance, the words "coração" heart and "panda" panda are masculine, despide their endings.

Words ending in -l and -e are very difficult to guess, so when learning these words, always remember their gender as well. Consider these, for instance:
 * 1) "sal" salt is masculine;
 * 2) "final" final can be either masculine or feminine, depending on the use:
 * 3) *When speaking of the ending of something, "final" is masculine; when this word refers to the last game on some championship or tournment, it is feminine.
 * 4) "cone" cone is masculine;
 * 5) "estudante" student can be either masculine or feminine.

title: Portuguese/Contents/Number and Gender


text:
User:Sytone

Hello World!

I am working on the C# page at the moment when I get a chance.

title: User:Sytone


text:
User:Korath

See w:User:Korath. (Cross-wiki redirects are evil.)

I'm not likely to do much here, other than fix the odd typo that catches my eye.

title: User:Korath


text:
Cookbook:Pickling

Pickling is the preservation of food in acid. It is used to preserve vegetables, fruit, fish, herbs, meat, and even hard-boiled eggs.

Pickling brine is a moderately strong acid which inhibits the growth of microbes; a high content of salt and/or sugar often enhances this effect. Some foods have a high water content, in which case methods must be used to ensure that the brine is strong enough to function properly.

Some of the more popular pickled foods include pickles (cucumbers), sauerkraut (cabbage), and green olives (olives).

These three examples cover the range of methods used to produce acidic pickling brine. In the case of cucumbers, vinegar (acetic acid) is usually added to provide all or most of the necessary acidity. Sauerkraut relies on a fermentation process to produce lactic acid, so careful control of temperature and salt content are crucial to the recipe. Fresh olives are naturally quite acidic, and so acid must be removed before they become palatable; depending upon the process used, several batches of brine might draw acid from the fruit and be thrown away before the final batch is added for storage.

It is widely believed that sushi has its origins in a method of pickling. Its defining ingredient is rice flavored with a brine of vinegar, sugar, and salt.

title: Cookbook:Pickling


text:
Portuguese/Practise Definite and Indefinite Articles/Answers

Answers
Answers to the lesson about the articles:


text:
Cookbook:Vinegar

Vinegar is a liquid consisting mainly of acetic acid (CH3COOH) and water. The acetic acid is produced by a process of fermentation of ethanol by acetic acid bacteria. Traditionally vinegar is made from a carbohydrate (fruit, grain, etc.) which can yield complex flavors, aromas, and colors. Commercial white vinegar, however, is basically a pure chemical.

Alcoholic drinks come in many basic forms, each defined by its ingredients and methods of manufacture; the same processes can be adjusted to produce acetic acid rather than ethanol, yielding a variety of vinegars. Some of these are named for their analogous alcohol.

pH
Vinegar has a pH value of about 3.

Density
Vinegar has a density of approximately 0.96 g/mL. The density level depends on the acidity of the vinegar.

Traditional balsamic vinegar
Balsamic vinegar is an aromatic aged vinegar produced in the Modena and Reggio Emilia provinces of Italy. The original product—Traditional Balsamic Vinegar—is made from the concentrated juice, or must of white Trebbiano grapes. It is very dark brown in color and its flavor is rich, sweet, and complex, with the finest grades being the product of years of aging in a successive number of casks made of various types of wood (including oak, mulberry, chestnut, cherry, juniper, ash, and acacia). Originally an artisanal product available only to the Italian upper classes, balsamic vinegar became widely known and available around the world in the late 20th century. True balsamic is aged at least 12 years, and is protected under the European Protected Designation of Origin (PDO) system. One can sometimes even find balsamic vinegars that have been aged for up to 100 years, though they are usually very expensive. Balsamic has a high acid level, but the sweetness covers the tart flavor, making it very mellow.

Balsamic wine vinegar
The commercial balsamic sold in supermarkets is typically made with red wine vinegar or concentrated grape juice mixed with a strong vinegar which is laced with caramel and sugar. However produced, balsamic needs to be made from a grape product. it is also black

Cider vinegar

 * Cider vinegar (apples)

Malt vinegar
Malt vinegar is made by malting barley, causing the starch in the grain to turn to maltose. An ale is then brewed from the maltose and allowed to turn into vinegar, which is then aged. It is typically light brown in color.

A cheaper alternative, called "non-brewed condiment," is a solution of 4-8% acetic acid colored with caramel (usually E150). There is also around 1-3% citric acid present.

Rice vinegar

 * Rice vinegar

Rice vinegar is most popular in the cuisines of East and Southeast Asia. It is available in white (actually light yellow), red, and black variants. In Japanese cuisine, a light and more delicate rice vinegar is preferred for the preparation of sushi rice and salad dressings. Red rice vinegar is traditionally colored with red yeast rice, although some Chinese brands use artificial food coloring instead. Black rice vinegar (made with black glutinous rice) is most popular in China, although it is also produced in Japan (see East Asian black, below). It may be used as a substitute for balsamic vinegar, although its dark color and the fact that it is aged may be the only similarity between the two products. Some varieties of rice vinegar are sweetened or otherwise seasoned with spices or other added flavorings.

White vinegar
White vinegar can be made by oxidizing a distilled alcohol. Alternatively, it may be nothing more than a solution of acetic acid and salt in water. Most commercial white vinegars are 5% acetic acid solutions, although some US states such as Virginia have laws prohibiting the sale as vinegar of any product not made from acetous fermentation of alcohol. These are made from alcoholic solutions fermented from cereals (often maize) and water.

Wine vinegar
Wine vinegar is made from red or white wine, and is the most commonly used vinegar in Mediterranean countries and Central Europe. As with wine, there is a considerable range in quality. Better quality wine vinegars are matured in wood for up to two years and exhibit a complex, mellow flavor. Wine vinegar tends to have a lower acidity than that of white or cider vinegars. There are more expensive wine vinegars made from individual varieties of wine, such as Champagne, Sherry, or Pinot Grigio.

title: Cookbook:Vinegar


text: Template:Portuguese

title: Template:Portuguese


text:
Cookbook:Cornflour

Cornflour is an ambiguous term that can mean cornstarch or finely ground cornmeal.

title: Cookbook:Cornflour


text:
Radiation Oncology/Anatomy

See also: Contouring

'''This page is for Anatomy Pearls useful for radiation oncology. Suitable examples include radiographic landmarks, location of lymph node groups, and lymph drainage patterns.'''

Anatomy atlases

 * HeadNeckBrainSpine Neuroradiology
 * Gray's Anatomy @ Bartleby.com
 * Basic Anatomy @ Dartmouth
 * Wesley Norman Anatomists
 * AnatomyAtlases.org
 * Radiology Assistant
 * Cross-sectional nodal atlas, 1999 (Ohio State) - PMID 10352611 Full text — "Cross-sectional nodal atlas: a tool for the definition of clinical target volumes in three-dimensional radiation therapy planning." Martinez-Monge R et al. Radiology. 1999 Jun;211(3):815-28.
 * Atlas of everything (H&N, lung, GI, GU, Gyn) on CT images with nodal regions shaded.
 * Wayne State University link
 * Brain: MRI (Sagittal, Coronal, Axial)
 * Upper Abdomen: CT
 * Thorax: CT
 * Pelvis + Upper Thigh: CT

Dissections

 * SUNY Downstate

Vertebral Levels
D means disc below level

Categorized: Uncategorized:
 * Vocal Cord —> C5
 * Cricoid —> C6
 * Start of trachea —> C6-C7
 * Thyroid gland —> C7
 * Lung apex —> T1
 * Origin SVC —> T3
 * AO arch —> T4/DT3
 * Carina —> T5
 * 25 cm from the incisors
 * Liver
 * Upper level: T7-8
 * Suprasternal Notch —> T3
 * Manubrium —> T4
 * Angle of louis —> T5
 * Xiphoid Process —> T10
 * Ant costodiaphragm —> T10
 * Post costodiaphragm —> L3
 * GEJ —> T10/T11 ( 2-3cm to left of midline )
 * 40cm from incisors
 * Spleen —> T10-T11 ( 5cm left to midline )
 * Pancreas body —> L1
 * Cysterna chili —> L1-L2
 * Umbilicus —> L4
 * If standing and in infants —> lower
 * Gallbladder —> at 9th rib; right lat border of rectus abdomens ( transpyloric plane )
 * Start of large bowel ( Cecum ) —> L4
 * Ascending Colon : 15cm
 * Descending Colon : 25cm
 * Rectum —>
 * Starts : S3
 * Ends : 4cm from anus
 * R Kidney hilum —> L2
 * L Kidney hilum —> L1
 * Para-AO nodes —> T12-L4
 * Common Iliac nodes —> L4 ( Bifurcation of AO )
 * Bifurcation of common iliac —> L5-S1
 * Hyoid Bone C4
 * Right Lung T1-T11
 * Left Lung T1-DT10
 * Suprasternal Notch T3
 * Aortic arch DT3
 * Origin SVC T3
 * Bifurcation of Trachea T5
 * SVC Heart junction T7
 * IVC Heart T9
 * Apex of Heart T10
 * Fundus stomach T10
 * Spleen T10-L1
 * GE Junction T11
 * Xiphoid Apex DT11
 * Left Kidney DT11-L3
 * Right Adrenal T11-L1
 * Left Adrenal T12-L1
 * Superior duodenum DT12
 * Colon splenic Flexure T12
 * Pancreas Body T12
 * Right lobe of Liver T9-DL3
 * Right Kidney T12-L3
 * Duodenal Jejeunal Flexure L1
 * Stomach pylorus L1
 * Colon hepatic flexure L2
 * Bottom of gallbladder L2
 * Aortic bifurcation L4
 * Iliac Crest L4
 * Umbilicus L4
 * Ileo Cecal Valve L5
 * Base of Appendix L5
 * IVC origin L5
 * Lower border of Cecum DS2

Clinical Syndromes with CN Palsy

 * ''Please see the Clinical Syndromes with CN Palsy page

Head & Neck

 * ''Please see the H&N anatomy page

Regional breast lymph nodes

 * ''Please see the axilla RT technique section

Supraclavicular
Brachial plexus:
 * Please see the Brachial plexus section
 * Also see the section on this page Radiation_Oncology/Anatomy

Thorax

 * See also: Radiation Oncology/Thorax/Anatomy (Lung lymph node stations)

Vessels
In order, superior to inferior:
 * Celiac artery
 * At T12 (75%) or L1 (25%)
 * SMA
 * Renal arteries/veins
 * Ovarian/testicular vein - Right side, drains into IVC 1 cm below R renal vein. Left side, drains into L renal vein lateral to the aorta.
 * Ovarian/testicular arteries - 5 to 6 cm above aortic bifurcation
 * IMA - 3 to 4 cm above aortic bifurcation

Lymphatics

 * Pancreaticoduodenal nodes - lie in C-loop of duodenum

 * Paraaortic lymph nodes (may also be called para-aortic, periaortic, or peri-aortic) - located adjacent to aorta, anterior to lumbar spine, extending bilaterally to the medial margins of the psoas major muscles, up to diaphragmatic crura.
 * For a paraaortic lymph node dissection, the surgeon usually dissects from aortic bifurcation up to SMA or renal veins.

CT Atlas

 * Baylor; 2007 PMID 17959923 -- "Learning the nodal stations in the abdomen." (Moron FE, Br J Radiol. 2007 Oct;80(958):841-8.)

Pelvic lymph nodes

 * Beth Israel, 2010 - PMID 20032141 -- "Lymphatic metastases from pelvic tumors: anatomic classification, characterization, and staging." (McMahon CJ, Radiology. 2010 Jan;254(1):31-46.)
 * Comment: shows subdivisions of lymph node groups. Normal node size cutoffs.
 * Park (MDACC), 1994 - PMID 7855343 — "Pathways of nodal metastasis from pelvic tumors: CT demonstration." (Park JM, Radiographics. 1994 Nov;14(6):1309-21.)

Prostate

 * Please see Radiation_Oncology/Prostate/Anatomy

Skin LN drainage

 * Auckland; 2007 (New Zealand) PMID 17643351 -- "Three-dimensional visualisation of lymphatic drainage patterns in patients with cutaneous melanoma." (Reynolds HM, Lancet Oncol. 2007 Sep;8(9):806-12.)
 * Heat maps based on 5239 SLN analyses of cutaneous melanoma patients
 * Conclusion: Sappey's lines not effective

Pterygopalatine ganglion

 * Suspended from V2 branch of trigeminal (CNV) nerve
 * Parasympathetic input from superior salivary nucleus in brain stem travel via facial nerve (CNVII) to greater superficial petrosal nerve
 * Sympathetic fibers from carotid plexus travel via deep petrosal nerve
 * The deep petrosal nerve and greater superficial petrosal nerves join to form the vidian nerve, which passes through (sympathetic fibers)/into (parasympathetic fibers) the pterygopalatine nucleus
 * Postganglionic parasympathetic fibers and passing-through sympathetic fibers travel through trigeminal nerve to lacrimal gland and nasal/oral mucosa

External Resources

 * Diagnostic Radiology
 * Nuclear Medicine

title: Radiation Oncology/Anatomy