To calculate the expected value of the proposition, we need to account for all possible outcomes and their respective probabilities.

Step 1: Determine the possible outcomes

You are tossing a coin three times, and there are 8 possible outcomes (since each toss has two possible results: heads (H) or tails (T)).

The possible outcomes for three coin tosses are:

• HHH
• HHT
• HTH
• HTT
• THH
• THT
• TTH
• TTT

Out of these, you win $55 if exactly two heads appear. The outcomes with exactly two heads are:

• HHT
• HTH
• THH

So, there are 3 favorable outcomes (two heads).

Step 2: Calculate the probabilities

• The probability of getting exactly two heads is the number of favorable outcomes (3) divided by the total number of outcomes (8), so: $P(\text{exactly 2 heads}) = \frac{3}{8}$

• The probability of not getting exactly two heads is the complement of the previous probability, so: $P(\text{not exactly 2 heads}) = 1 - P(\text{exactly 2 heads}) = 1 - \frac{3}{8} = \frac{5}{8}$

Step 3: Calculate the expected value

The expected value (EV) is calculated by multiplying each outcome by its probability and then summing those values:

$\text{EV} = \left(\text{Win Amount} \times P(\text{exactly 2 heads}))\right) + \left(\text{Loss Amount} \times P(\text{not exactly 2 heads}))\right)$

Substitute the values: $\text{EV} = (55 \times \frac{3}{8}) + (-28 \times \frac{5}{8})$ $\text{EV} = \frac{165}{8} - \frac{140}{8}$ $\text{EV} = \frac{25}{8}$ $\text{EV} = 3.125$

Step 4: Round the result

The expected value is $3.13 (rounded to two decimal places).

Final Answer:

The expected value of the proposition is $3.13.

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Would you like a more detailed explanation or have any questions about this process?

Here are some related questions for further exploration:

1. What is the probability of getting exactly 1 head in 3 tosses of a coin?
2. How do you calculate the expected value in a situation with multiple outcomes?
3. What happens to the expected value if the win amount increases?
4. How would you calculate the expected value for a game with more than two possible outcomes?
5. How can you use the expected value to determine whether a game is fair?

Tip: When calculating expected value, always ensure that you account for the probability of all possible outcomes and their respective payoffs, including any losses!