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: Calculate the probability of picking a diamond and not picking a diamond

In a standard deck of 52 cards, there are 13 diamonds. So the probability of picking a diamond, \(P(D)=\frac{13}{52}=\frac{1}{4}\). The probability of not picking a diamond, \(P(
eg D)=1 - \frac{1}{4}=\frac{3}{4}\).

Step2: Set up the expected - value formula for a fair game

For a fair game, the expected value \(E(X)=0\). Let \(x\) be the number of points for picking a diamond. The expected - value formula is \(E(X)=x\times P(D)+(- 2)\times P(
eg D)\).
Substitute the probability values into the formula: \(0 = x\times\frac{1}{4}+(-2)\times\frac{3}{4}\).

Step3: Solve the equation for \(x\)

First, simplify the right - hand side of the equation: \(0=\frac{x}{4}-\frac{6}{4}\).
Add \(\frac{6}{4}\) to both sides of the equation: \(\frac{x}{4}=\frac{6}{4}\).
Multiply both sides by 4 to solve for \(x\): \(x = 6\).

Answer:

6