To solve this problem, we need to calculate the expected value (expected winnings) based on the outcomes of tossing three fair coins and the respective payouts for each possible number of heads.

Step 1: Determine Probabilities

When tossing three fair coins, the possible outcomes for the number of heads can range from 0 to 3. Let's calculate the probability for each outcome:

1. 3 heads: There is only 1 way to get all heads, out of 8 possible outcomes (HHH).

   • Probability of 3 heads = $\frac{1}{8}$

2. 2 heads: There are 3 ways to get exactly 2 heads (HHT, HTH, THH).

   • Probability of 2 heads = $\frac{3}{8}$

3. 1 head: There are 3 ways to get exactly 1 head (HTT, THT, TTH).

   • Probability of 1 head = $\frac{3}{8}$

4. 0 heads: There is only 1 way to get no heads, which is all tails (TTT).

   • Probability of 0 heads = $\frac{1}{8}$

Step 2: Calculate Expected Winnings

The expected winnings are calculated by multiplying each outcome's payout by its probability and summing these products.

The payouts for each outcome are:

• 3 heads: 23 cents
• 2 heads: 8 cents
• 1 head: 4 cents
• 0 heads: 0 cents

So, the expected winnings $E$ can be calculated as: $E = (23 \times \frac{1}{8}) + (8 \times \frac{3}{8}) + (4 \times \frac{3}{8}) + (0 \times \frac{1}{8})$

Calculating each term:

1. 23 cents for 3 heads: $23 \times \frac{1}{8} = 2.875$
2. 8 cents for 2 heads: $8 \times \frac{3}{8} = 3$
3. 4 cents for 1 head: $4 \times \frac{3}{8} = 1.5$
4. 0 cents for 0 heads: $0 \times \frac{1}{8} = 0$

Adding these up: $E = 2.875 + 3 + 1.5 + 0 = 7.375 \text{ cents}$

Step 3: Determine if 8 Cents is a Fair Price

A fair price would be one where the cost to play equals the expected winnings. Here, the expected winnings are 7.375 cents, which is less than the 8 cents required to play.

Thus:

• The 8-cent cost to play is not a fair price because the expected winnings are only 7.375 cents.

Would you like more details on any part of this solution, or do you have additional questions?

Here are five related questions to explore further:

1. How would the game change if the payout for 3 heads increased?
2. What would be the fair price if the payouts were modified?
3. How does the probability of each outcome affect expected winnings?
4. What if the game used four coins instead of three? How would that impact payouts?
5. What is the role of variance in determining whether a game is fair?

Tip: In probability, calculating expected values helps determine fair pricing in games of chance by assessing average winnings over many trials.