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Problem 687

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Shuffling Cards

A standard deck of $52$ playing cards, which consists of thirteen ranks (Ace, Two, …, Ten, King, Queen and Jack) each in four suits (Clubs, Diamonds, Hearts and Spades), is randomly shuffled. Let us call a rank perfect if no two cards of that same rank appear next to each other after the shuffle.

It can be seen that the expected number of ranks that are perfect after a random shuffle equals $\frac {4324} {425} \approx 10.1741176471$.

Find the probability that the number of perfect ranks is prime. Give your answer rounded to $10$ decimal places.

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洗牌

一副拿走大小王的标准扑克牌有$52$张牌，分为四种花色（草花、方块、红心、黑桃）和十三套点数（$A$，$2$，……，$10$，$J$，$Q$，$K$）。在洗牌之后，如果某套点数的四张牌相互之间都不紧挨着，则称这套点数被完美洗开了。

可以计算出，洗一次牌之后，被完美洗开的点数的期望是$\frac {4324} {425} \approx 10.1741176471$套。

求被完美洗开的点数恰好有质数套的概率，并将你的答案保留小数点后$10$位。

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