A deck holds $83$ cards carrying the distinct numbers $1$ through $83$. After a random shuffle, the first $5$ cards are dealt. It costs $1 to play, and you receive $x$ if those five cards come out in strictly increasing or strictly decreasing order (and nothing otherwise). For which $x$ is the game fair?

Fair prize for five cards dealt in sorted order

Question

A deck holds $83$ cards carrying the distinct numbers $1$ through $83$ . After a random shuffle, the first $5$ cards are dealt. It costs $1 to play, and you receive \text{\char36} $x$ if those five cards come out in strictly increasing or strictly decreasing order (and nothing otherwise). For which $x$ is the game fair?

Short answer

$x=60$ . Only the relative order of the five dealt cards matters, and all $5!=120$ orders are equally likely. Two of them are monotone, so you win with probability $\tfrac1{60}$ , and a fair prize is $60$ dollars.

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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