Problem:
Each of 20 balls is tossed independently and at random into one of 5 bins. Let be the probability that some bin ends up with 3 balls, another with 5 balls, and the other three with 4 balls each. Let be the probability that every bin ends up with 4 balls. What is ?
Answer Choices:
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Solution:
The requested ratio divides the number of ways to end up with a 3-4-4-4-5 distribution by the number of ways to end up with a distribution. For either outcome, there are at least three bins with 4 balls each, leaving 8 balls to distribute into two bins. For a split in the two bins, there are ways to choose the bins, and ways to choose 3 balls. For a split in the two bins, there are ways to choose 4 balls. The requested ratio is therefore .
The probabilities of the ball distribution follow the multinomial distribution. If there are balls and bins, then the probability that balls end up in bin for every is given by
where is the probability of any particular ball getting tossed into bin , which equals . There are 5.4 choices for which bin gets 3 balls and which bin gets 5 balls under the first scenario. The requested ratio is therefore
The problems on this page are the property of the MAA's American Mathematics Competitions