Question

A game is played with a deck of cards. In this game, if a player picks a diamond, he/she will win $20. If he/she picks a face card, which is not a diamond, he/she will win $10. If he/she picks any other card, he/she will lose $7. Check the fairness of the game. The game is not fair, but favorable to the player.The game is fair, but not favorable to the player.The game is fair.The game is fair and favorable to the player.

Answer 1

The game is not fair, but favorable to the player.

Explanation:

In a standard deck, there are 52 cards.

There are 12 face cards (Kings, queens, and jacks) in a deck (3 for each suit).

• The number of diamonds = 13

,

• The number of face cards (which is not a diamond) = 9

,

• The number of other cards = 52-(13+9)=30

The respective probabilities are given below:

`\begin{gathered} P(\text{diamonds)}=(13)/(52) \\ P(\text{face cards which is not a diamond)}=(9)/(52) \\ P(\text{other cards)}=(30)/(52) \end{gathered}`

To determine if the game is fair, we find the expected value.

Let X be the profit in each case.

The table below describes the probability distribution of X.

Thus:

`\begin{gathered} \text{Expected Value of X, E(X)}=\sum xP(x) \\ =(20*(13)/(52))+(10*(9)/(52))+(-7*(30)/(52)) \\ =\$2.69 \end{gathered}`

The game always results in a profit for the player. Thus, the game is not fair.

The game is not fair, but favorable to the player.