Question

A card game using a standard deck of 52 cards costs $1 to play. If you draw a face card or an even number, you lose. If you draw an ace, you win $5, and if you draw an odd number, you win $2. What is the expected payoff?

If the game is not fair, who has the advantage? Explain your reasoning.

Answer 1

The anticipated winnings is $1 and the cost to play is $1, so the expected payoff is $0.

Explanation:

The payoff of the card game is the difference between the cost to play ($1) and the average probable winnings.

The deck being used has 13 ordinal cards: One (also known as the Ace) through 10, plus the Face Cards: Jack, Queen, King. There are 4 suits (spades, clubs, hearts, diamonds): `13xx4=52`.

Out of the 52 cards that can be drawn, you get nothing if you draw a face card or an even number. Let's figure out how many cards that is:

2, 4, 6, 8, 10, J, Q, K = 8 cards in 4 suits = `8xx4=32` cards with a winning value of $0.

You get $5 if you draw an Ace, so that's 4 cards with value $5

You get $2 if you draw an odd card, so that's 3, 5, 7, 9 = 4 cards in 4 suits: `4xx4=16` cards with winning value $2.

So that makes the overall average probable winning value:

`(32/52)xx0+(4/52)xx5+(16/52)xx2=20/52+32/52=52/52=1`

The expected payoff is therefore:

`$1-$1=$0`