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?

Explanation:

Step1: Calculate probability of getting two tails

When flipping two coins, total outcomes \(n(S)=2\times2 = 4\) (HH, HT, TH, TT). Probability of getting two tails \(P(TT)=\frac{1}{4}\)

Step2: Calculate probability of non - two - tails outcomes

Probability of non - two - tails outcomes \(P(\text{not }TT)=1 - P(TT)=1-\frac{1}{4}=\frac{3}{4}\)

Step3: Set up the expected value formula for a fair game

Let \(x\) be the number of points lost for non - two - tails outcomes. Expected value \(E(X)\) of a game is given by \(E(X)=\sum_{i}x_ip_i\). For a fair game \(E(X) = 0\).
\(E(X)=(1)\times\frac{1}{4}+(-x)\times\frac{3}{4}=0\)

Step4: Solve the equation for \(x\)

$$
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$$

Answer:

\(\frac{1}{3}\)