you pick a card at random from an ordinary deck of 52 cards. if the card is a diamond, you get some points…

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?\n12\n6\n24\n3

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?\n12\n6\n24\n3

Answer

Explanation:

Step1: Calculate probabilities

The probability of drawing a diamond, $P(D)=\frac{13}{52}=\frac{1}{4}$, and the probability of not - drawing a diamond, $P(\overline{D})=\frac{39}{52}=\frac{3}{4}$.

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

Let the value of getting a diamond be $x$ points. For a fair game, the expected value $E(X) = 0$. The expected - value formula 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 $x$, so $x = 6$.

Answer:

B. 6