Question

you pick a card at random from an ordinary deck of 52 cards. if the card is a diamond, you get some points; if not, you lose 2 points. what value for the diamonds would make the game fair?
12
6
24
3

Explanation:

Step1: Define Probabilities

A deck has 52 cards, 13 diamonds. Probability of diamond ($P(D)$) is $\frac{13}{52}=\frac{1}{4}$. Probability of non - diamond ($P(
eg D)$) is $\frac{39}{52}=\frac{3}{4}$.

Step2: Fair Game Condition

For a fair game, expected value $E = 0$. Let $x$ be points for diamond. $E=P(D)\times x+P(
eg D)\times(- 2)=0$.

Step3: Solve for x

Substitute probabilities: $\frac{1}{4}x+\frac{3}{4}\times(-2)=0$. Multiply through by 4: $x - 6 = 0$. So $x = 6$.

Answer:

6