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?
o 12
o 6
o 24
o 3

Explanation:

Step1: Calculate probabilities

There are 13 diamonds in a 52 - card deck, so the probability of picking a diamond $P(D)=\frac{13}{52}=\frac{1}{4}$, and the probability of not picking a diamond $P(\overline{D}) = 1-\frac{1}{4}=\frac{3}{4}$.

Step2: Set up expected - value equation for a fair game

Let the value of getting a diamond be $x$. The value of not getting a diamond is - 2. For a fair game, the expected value $E(X)=0$. The formula for expected value is $E(X)=P(D)\times x+P(\overline{D})\times(- 2)$. Substituting the probabilities, we get $0=\frac{1}{4}x+\frac{3}{4}\times(-2)$.

Step3: Solve the equation for $x$

First, expand the right - hand side of the equation: $0=\frac{1}{4}x-\frac{6}{4}$. Then, add $\frac{6}{4}$ to both sides: $\frac{6}{4}=\frac{1}{4}x$. Multiply both sides by 4 to solve for $x$: $x = 6$.

Answer:

6