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HardProbability Combinatorics~9m

Seven people each independently pick a uniformly random integer from $\{1, 2, \dots, 7\}$ . Given that nobody picked $7$ , what is the probability that $7$ is the only value nobody picked (i.e. every one of $1, \dots, 6$ was chosen by someone)?

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Given nobody picked $7$ , all $7$ picks are uniform on $\{1,\dots,6\}$ ( $6^7$ sequences).
" $7$ the only missing value" means the $7$ picks are onto $\{1,\dots,6\}$ : $6!\,S(7,6)=6!\binom72$ surjections.

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0.054 (± 0.001)

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Asked at: Jane Street, Two Sigma

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