Suppose you enter a contest where your probability of coming in first place is 0.27 and your probability of coming in second place is 0.36. If you come in first place, you win $70. If you come in second place, you win $36. If you don't come in first or second place, you have to pay $55. What is the expected value of this game?

To calculate the expected value of this game, we need to consider the probabilities of each outcome along with the corresponding monetary outcomes. The expected value (EV) is calculated as follows:

\[
\text{EV} = (\text{Probability of Winning 1st Place} \times \text{Prize for 1st Place}) + (\text{Probability of Winning 2nd Place} \times \text{Prize for 2nd Place}) + (\text{Probability of Not Placing} \times \text{Loss})
\]

Given:
- Probability of coming in 1st place: $ 0.27 $
- Prize for 1st place: $ \$70 $
- Probability of coming in 2nd place: $ 0.36 $
- Prize for 2nd place: $ \$36 $
- Probability of not placing (not 1st or 2nd): $ 1 - (0.27 + 0.36) = 1 - 0.63 = 0.37 $
- Loss for not placing: $ -\$55 $

Now, we calculate the expected value:

\[
\text{EV} = (0.27 \times 70) + (0.36 \times 36) + (0.37 \times -55)
\]

Calculating each term:

\[
0.27 \times 70 = 18.9
\]
\[
0.36 \times 36 = 12.96
\]
\[
0.37 \times -55 = -20.35
\]

Now, sum these values:

\[
\text{EV} = 18.9 + 12.96 - 20.35 = 11.51
\]

Thus, the expected value of the game is **\$11.51**.

Would you like more details or have any questions?

Here are five related questions to consider:
1. How would the expected value change if the prize for first place increased to \$80?
2. What would happen to the expected value if the probability of not placing increased to 0.40?
3. If the entry fee for the contest were \$10, what would the net expected value be?
4. How does the risk of paying \$55 affect the desirability of entering this contest?
5. What is the probability threshold where the game no longer has a positive expected value?

**Tip:** The expected value helps you understand the average outcome over many trials, but remember that it doesn't guarantee a win or loss in any single trial.

Learn how to calculate the expected value of a contest where probabilities of winning first and second place, along with monetary outcomes, are given. This problem involves probability theory and financial calculations.

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