Question

you play a game in which two coins are flipped. if both coins turn up tails, you win 1 point. how many points would you need to lose for each of the other outcomes so that the game is fair?
1/4
1/3
1
4/3
done

Explanation:

Step1: Calculate total number of outcomes

When two coins are flipped, the total number of possible outcomes is $2\times2 = 4$ (HH, HT, TH, TT).

Step2: Determine probability of winning

The probability of getting two - tails (TT) is $P(\text{win})=\frac{1}{4}$, and the probability of losing is $P(\text{lose})=\frac{3}{4}$.

Step3: Set up expected - value equation for a fair game

Let $x$ be the number of points lost for non - TT outcomes. For a fair game, the expected value $E(X)=0$. The expected - value formula is $E(X)=P(\text{win})\times1+P(\text{lose})\times(-x)$. Substituting the probabilities, we have $0=\frac{1}{4}\times1+\frac{3}{4}\times(-x)$.

Step4: Solve the equation for $x$

First, expand the equation: $0 = \frac{1}{4}-\frac{3x}{4}$. Then, add $\frac{3x}{4}$ to both sides: $\frac{3x}{4}=\frac{1}{4}$. Multiply both sides by $\frac{4}{3}$ to get $x=\frac{1}{3}$.

Answer:

$\frac{1}{3}$