To calculate the expected value of this game, we need to compute the weighted average of the payouts, where each outcome is weighted by its probability.

Step 1: Identify the outcomes and their probabilities

When flipping two coins, the possible outcomes are:

1. Both coins show tails (TT):

   • Probability = $\frac{1}{4}$ (since $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$).

2. One coin shows heads and the other shows tails (HT or TH):

   • Probability = $\frac{1}{2}$ (since there are two outcomes, each with $\frac{1}{4}$ probability).

3. Both coins show heads (HH):

   • Probability = $\frac{1}{4}$.

Step 2: Define the payouts

• For TT: You win $9.
• For HT or TH: You lose $3.
• For HH: No money is exchanged, so the payout is $0.

Step 3: Compute the expected value

The expected value (EV) is calculated using the formula: $\text{EV} = (\text{Payout for TT} \times \text{Probability of TT}) + (\text{Payout for HT/TH} \times \text{Probability of HT/TH}) + (\text{Payout for HH} \times \text{Probability of HH})$

Substitute the values: $\text{EV} = (9 \times \frac{1}{4}) + (-3 \times \frac{1}{2}) + (0 \times \frac{1}{4})$

Simplify: $\text{EV} = \frac{9}{4} + \left(-\frac{3}{2}\right) + 0$

Convert to a common denominator ($4$): $\text{EV} = \frac{9}{4} - \frac{6}{4} = \frac{3}{4}$

Step 4: Interpret the result

The expected value of the game is: $\text{EV} = \frac{3}{4} \text{ dollars} = 75 \text{ cents}$

This means, on average, you can expect to gain 75 cents per game.

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Do you want details on how probabilities or expectations are calculated? Here are some related questions for exploration:

1. How does changing the payouts affect the expected value?
2. What if the coins are biased? How would probabilities change?
3. How does the law of large numbers apply to repeated games like this?
4. Can we design a fair game with adjusted payouts?
5. What is the variance of the payouts in this game?

Tip: To test your understanding of expected value, try calculating it for games with more complex probabilities or payouts!