you play a game in which two coins are flippe...
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 $\frac{1}{3}$ $\frac{1}{4}$ $\frac{1}{2}$
Answer
# 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 $P(TT)=\frac{1}{4}$.
## Step3: Determine probability of losing
The probability of not getting two - tails is $P(\text{not TT}) = 1-\frac{1}{4}=\frac{3}{4}$.
## Step4: 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 $E(X)=1\times P(TT)+(-x)\times P(\text{not TT})$. Substituting the probabilities, we get $0 = 1\times\frac{1}{4}+(-x)\times\frac{3}{4}$.
## Step5: Solve the equation for $x$
\[
\begin{align*}
0&=\frac{1}{4}-\frac{3x}{4}\\
\frac{3x}{4}&=\frac{1}{4}\\
x&=\frac{1}{3}
\end{align*}
\]
# Answer:
$\frac{1}{3}$