Answer:
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:
[tex]\begin{gathered} P(\text{diamonds)}=\frac{13}{52} \\ P(\text{face cards which is not a diamond)}=\frac{9}{52} \\ P(\text{other cards)}=\frac{30}{52} \end{gathered}[/tex]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:
[tex]\begin{gathered} \text{Expected Value of X, E(X)}=\sum xP(x) \\ =(20\times\frac{13}{52})+(10\times\frac{9}{52})+(-7\times\frac{30}{52}) \\ =\$2.69 \end{gathered}[/tex]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.
