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 $\frac{1}{17}$$\frac{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.

856. 等待一对牌

一副标准的 52 张扑克牌有 4 种花色，每种花色都有 13 张不同点数的牌。两张点数相同牌被称为一个 对子。

我们将从一副洗好的牌中，不放回地逐张摸牌，直至有连续的两张牌形成对子或者牌堆已空。例如，前两次摸出的牌恰成一对的概率是 $\frac{1}{17}$$\frac{1}{17}$。

求摸出的牌数量的期望。若将牌堆取空仍未有连续的一对，此时我们认为共摸出了 52 张牌。将答案四舍五入至小数点后第 8 位。

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