Distribution of The Balls
If a ball bounces to the right k times on its way down (and to the left on the remaining pins) it ends up in the kth bin counting from the left. Denoting the number of rows of pins in a bean machine by n, the number of paths to the kth bin on the bottom is given by the binomial coefficient . If the probability of bouncing right on a pin is p (which equals 0.5 on an unbiased machine) the probability that the ball ends up in the kth bin equals . This is the probability mass function of a binomial distribution.
According to the central limit theorem the binomial distribution approximates the normal distribution provided that n, the number of rows of pins in the machine, is large.
Read more about this topic: Bean Machine
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