84 Cards

In a game, there is a deck of 84 distinct cards, each labeled with a value from 1 to 84. A dealer shuffles the deck and deals the top 5 cards. If these top 5 cards are in strictly ascending or descending order, the player wins a payout of $x; otherwise, they win nothing. The cost to play the game is $1. What should the value of $x be for the game to be fair?

Answer

The total number of ways to choose 5 cards from 84 is:

\binom{84}{5}.

For any given 5 cards, there are 2 orders (strictly ascending or descending) out of $$5! = 120$$ possible orders. Thus, the probability of winning is:

\frac{2}{5!} = \frac{2}{120} = \frac{1}{60}.

For a fair game, the expected net payoff should be 0:

0 = \frac{1}{60}(x - 1) + \left(-1\right)\left(\frac{59}{60}\right).

Solving for $$x$$ gives:

\[ x = 60. \]