Chance operations & probability theory
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In this example a ball is dropped on the top pin of a triangular arrangement of pins. The ball ends up exiting in one of several columns. The probability that a ball will end up in a given column is a function of the number of paths that lead to that column. | In this example a ball is dropped on the top pin of a triangular arrangement of pins. The ball ends up exiting in one of several columns. The probability that a ball will end up in a given column is a function of the number of paths that lead to that column. | ||
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| + | <SPAN STYLE="font-size: larger;"><u>Pascal's Triangle</u></span> | ||
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| + | This geometric arrangement of numbers was devised by Blaise Pascal to demonstrate and calculate the probability of a proposition that is the combination of binary events. To construct Pascal's Triangle each number is the sum of the 2 numbers in the line above. The resulting numbers can be viewed as the expected statistical distribution. For example, flipping 3 coins there is only one way to get all heads or all tails, but 3 ways to get 1 tail and 2 heads (THH, HTH, HHT), and 3 ways to get 1 head and 2 tails. | ||
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| + | 1 | ||
| + | 1 1 | ||
| + | 1 2 1 | ||
| + | 1 3 3 1 | ||
| + | 1 4 6 4 1 | ||
| + | 1 5 10 10 5 1 | ||
| + | 1 6 15 20 15 6 1 | ||
| + | 1 7 21 35 35 21 7 1 | ||
| + | 1 8 28 56 70 56 28 8 1 | ||
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| + | To calculate the probability of a given outcome one simply takes the number of combinations that can lead to the desired result, divided by the number of possible results. So the probability of flipping 3 coins and getting a head and 2 tails is: | ||
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| + | (3) / ( 1 + 3 + 3 + 1 ) = 3/8 = .375 | ||
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| + | The probability of flipping 6 coins and getting the most likely result of 3 heads and 3 tails is: | ||
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| + | (20) / ( 1 + 6 + 15 + 20 + 15 + 6 + 1 ) = 20/64 = .3125 | ||
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| + | Note that the result common sense would expect, an equal number of heads and tails, when flipping 6 coins happens less than than 1/3 of the time! | ||
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| + | <SPAN STYLE="font-size: larger;"><u>The Independence of Events and the Gambler's Fallacy</u></span> | ||
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| + | It's important to remember that each random event is unrelated to any other preceding event. While flipping 10 heads in a row is very unlikely, once one has flipped 9 heads in a row the probability for the 10th flip is the same as it ever was, .5 for heads and .5 for tails. Gamblers often think that so called "streaks" are meaningful, but in the case of random events they are not. Oddly, for every misguided gambler who thinks that 9 heads in a row constitutes a streak that is likely to continue, there is usually another gambler who thinks 9 heads in a row means that tails are "due". Both are wrong. | ||
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| + | <SPAN STYLE="font-size: larger;">The Normal or Gaussian Distribution</span> | ||
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| + | As more and more lines are added to Pascal's Triangle the distribution approximates a "bell curve" or the normal distribution, also known as a Gaussian Distribution. This distribution is found throughout nature, for example height and weight measurements of a given population will approximate a normal distribution. | ||
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| + | {{SingleImage|imageWidthPlusTen=594|imageURL=http://www-viz.tamu.edu/courses/viza658/wiki/prob/01.jpg|caption=Gaussian Distribution}} | ||
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| + | {{SingleImage|imageWidthPlusTen=730|imageURL=http://www-viz.tamu.edu/courses/viza658/wiki/prob/02.jpg|caption=Faces generated with psuedo-random numbers}} | ||
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| + | {{SingleImage|imageWidthPlusTen=730|imageURL=http://www-viz.tamu.edu/courses/viza658/wiki/prob/03.jpg|caption=Faces generated with Gaussian Distribution}} | ||
<SPAN STYLE="font-size: larger;"><u>The Law of Large Numbers</u></SPAN> | <SPAN STYLE="font-size: larger;"><u>The Law of Large Numbers</u></SPAN> | ||
Even though events in nature will often tend towards a normal distribution, any particular set of measures or samples will show variation. Where sample sizes are small the general distribution in nature may not be at all apparent. As the sample size grows the apparent distribution will incrementally grow closer to the actual distribution in all of nature. | Even though events in nature will often tend towards a normal distribution, any particular set of measures or samples will show variation. Where sample sizes are small the general distribution in nature may not be at all apparent. As the sample size grows the apparent distribution will incrementally grow closer to the actual distribution in all of nature. | ||
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| + | Here is a simulation of a pinball machine that demonstrates the Gaussian distribution of the result. Note how as more balls are put into play the distribution moves closer and closer to a Gaussian distribution. | ||
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| + | <a href="http://www.jcu.edu/math/isep/Quincunx/Quincunx.html">http://www.jcu.edu/math/isep/Quincunx/Quincunx.html</a> | ||
== Links == | == Links == | ||
