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?\no 12\no 6\no 24\no 3

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?\no 12\no 6\no 24\no 3

Answer

Answer:

A. 6

Explanation:

Step1: Calculate probabilities

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 number of points for picking a diamond be $x$. The expected value $E(X)$ of a fair game is 0. The formula for expected value is $E(X)=x\times P(D)+(- 2)\times P(\overline{D})$. So, $0 = x\times\frac{1}{4}+(-2)\times\frac{3}{4}$.

Step3: Solve the equation for $x$

First, expand the equation: $0=\frac{x}{4}-\frac{6}{4}$. Then, add $\frac{6}{4}$ to both sides: $\frac{x}{4}=\frac{6}{4}$. Multiply both sides by 4 to get $x = 6$.