Question

this problem is known as the banach match problem. a pipe-smoking mathematician always carries two matchboxes, one in his right-hand pocket and one in his left-hand pocket. each time he needs a match he is equally likely to take it from either pocket. the mathematician discovers that one of his matchboxes is empty. if it is assumed that both matchboxes initially contained n matches, what is the probability that there are exactly k matches in the other box, k

Answer 1

The Banach Match Problem involves using combinatorial probability to determine the likelihood of a certain number of matches remaining, while the other listed problems highlight different probability distributions such as hypergeometric, geometric, and conditional probabilities.

Step-by-step explanation:

The Banach Match Problem is an interesting probability puzzle. To calculate the probability that there are exactly k matches left in the other box after finding one empty, one would use combinatorial probability theory.

Assuming both matchboxes initially contained n matches, and matches are drawn with equal likelihood from either box, the number of ways to distribute the remaining matches can be obtained using combinations.

The specific formula would depend on carefully setting up the problem considering all possible distributions of the taken matches.

As for the problems listed, they are examples of different kinds of probability problems. Some exhibit characteristics of a hypergeometric distribution, where you deal with a finite population without replacement, some are examples of a geometric distribution, where you have a certain probability of success in each trial, and others are related to conditional probability and goodness-of-fit tests.