Question

A pipe-smoking mathematician always carries two boxes of matches - one is his right pocket and another in his left pocket. Each time he needs a match, he is equally likely to take it from either pocket. Suppose that each box initially contains n matches. What is the probability that once the mathematician discovers that one of the boxes is empty, there are exactly k matches in the other box, k = 0,1,...n?

Answer 1

$2\ variants\,\,to\,\,choose\,\,box\,\,which\,\,is\,\,found\,\,empty\\C_{n+k}^{k}\,\,variants\,\,for\,\,the\,\,sequence\,\,of\,\,matches\,\,from\,\,the\,\,boxes\\2^{n+k+1}\,\,total\,\,variants\,\,to\,\,choose\,\,a\,\,box\,\,\left( the\,\,last\,\,box\,\,is\,\,chosen\,\,that\,\,empty \right) \\P=\frac{2\cdot C_{n+k}^{k}}{2^{n+k+1}}=\frac{C_{n+k}^{k}}{2^{n+k}}$