Question

in a card game with my friend, I pay a certain amount of money each time I lose. i win $4 if I draw a jack or a queen and I win $5 if I draw a king or ace for an ordinary pack of 52 playing cards. if I draw other cards I lose. what should I pay so that we come out even? (that is if the game is fair).

Answer 1

Answer on Question #62619 – Math – Statistics and Probability

Question

In a card game with my friend, I pay a certain amount of money each time I lose.

I win $4 if I draw a jack or a queen and I win $5 if I draw a king or ace for an ordinary pack of 52 playing cards.

If I draw other cards I lose.

What should I pay so that we come out even? (That is if the game is fair).

Solution

The probability mass function of amount of money is as follows

The expectation (the expected value) of a random variable $X$ is equal to

$$E(X) = \sum x_i P(x_i) = 4 \times \frac{2}{13} + 5 \times \frac{2}{13} - x \times \frac{9}{13} = \frac{18 - 9x}{13}.$$

Now the expected value should be $0 for the game to be fair.

$$\frac{18 - 9x}{13} = 0 \Rightarrow x = 2.$$

So I should pay $2 for it to be a fair game.

Answer: $2.

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