Question

A box contain m white balls and n black balls. balls are drawn at random one at a time without replacement. find the probability of encountering a white ball at kth draw.

Answer 1

The probability of encountering a white ball by the kthkth is equal to 1−P1−P where PP is the probability that the first kk balls were all black. The latter probability is easier to compute. For example, the probability of the first ball being black is nm+nnm+n. The probability that the second ball is also black is n−1m+n−1n−1m+n−1. So, the probability that the first two balls are both black is n(n−1)(m+n)(m+n−1)n(n−1)(m+n)(m+n−1). The probability that the first kk balls are all black is similarly given by n(n−1)...(n−k+1)(m+n)(m+n−1)...(m+n−k+1)n(n−1)...(n−k+1)(m+n)(m+n−1)...(m+n−k+1).

Note that if k>nk>n, then the probability becomes 1 because clearly you will have to draw a white ball by the kthkth draw (a version of Pigeon Hole Principle