The question in the image seems to be asking whether a 3¢ cost to play a game involving tossing 3 fair coins is a fair price, based on expected winnings. The payouts for the game are structured as follows:

• 9¢ for 3 heads,
• 3¢ for 2 heads,
• 1¢ for 1 head.

We need to calculate the expected winnings and compare them to the 3¢ cost to determine if it's a fair price to pay.

Step 1: Probabilities of Each Outcome

For tossing 3 fair coins, the outcomes can be summarized using binomial probabilities:

• Probability of 3 heads: $\frac{1}{8}$ (since there is only 1 way to get all 3 heads out of 8 total outcomes),
• Probability of 2 heads: $\frac{3}{8}$ (since there are 3 ways to get exactly 2 heads),
• Probability of 1 head: $\frac{3}{8}$ (since there are 3 ways to get exactly 1 head),
• Probability of 0 heads: $\frac{1}{8}$, but since there is no payout for 0 heads, we ignore this case for calculating the expected value.

Step 2: Expected Value Calculation

The expected value (EV) is the sum of each outcome's value multiplied by its probability. So, the EV is:

$EV = \left(\frac{1}{8} \times 9\right) + \left(\frac{3}{8} \times 3\right) + \left(\frac{3}{8} \times 1\right)$

Simplifying this:

$EV = \frac{9}{8} + \frac{9}{8} + \frac{3}{8} = \frac{21}{8} = 2.625 \text{¢}$

Step 3: Conclusion

The expected winnings are 2.625¢. Since the cost to play is 3¢, which is higher than the expected value, the game is not fair. A fair price to pay would be equal to the expected winnings, which is 2.625¢.

Would you like more details on any part of this explanation?

Here are 5 related questions to expand on this:

1. What does "fair price" mean in terms of expected value in probability theory?
2. How do you calculate probabilities using binomial distributions?
3. What are other ways to adjust the game to make it fair?
4. Can we apply the same principles to other types of games or bets?
5. How does the law of large numbers relate to the concept of fairness in games?

Tip: When evaluating fairness in games, always compare the expected value with the cost to play. If the cost exceeds the expected value, the game is unfavorable for the player.