Problem 856

Waiting for a Pair

A standard $52$-card deck comprises $13$ ranks in four suits. A pair is a set of two cards of the same rank.

Cards are drawn, without replacement, from a well shuffled $52$-card deck waiting for consecutive cards that form a pair. For example, the probability of finding a pair in the first two draws is $1/17$.

Cards are drawn until either such a pair is found or the pack is exhausted waiting for one. In the latter case we say that all $52$ cards were drawn.

Find the expected number of cards that were drawn. Give your answer rounded to eight places after the decimal point.

等待对子

一副标准的$52$张扑克牌由四个花色各$13$个点数的牌组成。对子是指两张点数相同的牌。

从一副洗好的扑克牌中无放回地抽牌，目标是连续抽出的两张牌组成对子。例如，前两次抽牌就恰好抽出对子的概率是$1/17$。

持续抽牌直到抽出对子或所有的牌都已抽完，在后一种情况下抽牌数记为$52$。

求停止抽牌时抽牌数的期望值，并四舍五入保留八位小数作为你的答案。