Carnival Card Game

Solution

There are \(4\) aces in the deck. You win \$10 for each ace among the \(10\) cards, but if there are no aces you win \$0. The easiest way to evaluate this game is to compute the expected number of aces among the \(10\) cards. Each individual card has probability \[ \frac{4}{52} = \frac{1}{13} \] of being an ace. Since \(10\) cards are dealt, the expected number of aces is \[ 10 \cdot \frac{1}{13} = \frac{10}{13} \approx 0.769. \] Each ace pays \$10, so the expected payout is \[ 10 \times \frac{10}{13} = \frac{100}{13} \approx 7.69. \] Now subtract the \$5 entry cost: \[ 7.69 - 5 = 2.69. \] So your expected profit is about \[ \$2.69. \] Even though sometimes you will get zero aces and lose everything, the average payout from the aces you do get more than compensates for the \$5 entry fee. So the expected value of playing is positive, meaning you should play this game.