A local club plans to invest $10000 to host a baseball game. They expect to sell tickets worth $23567. But if it rains on the day of game, they won't sell any tickets and the club will lose all the money invested. If the weather forecast for the day of game is 20% possibility of rain,what is the expected value of investing in the baseball game?

Question

A local club plans to invest $10000 to host a baseball game. They expect to sell tickets worth $23567. But if it rains on the day of game, they won't sell any tickets and the club will lose all the money invested. If the weather forecast for the day of game is 20% possibility of rain,what is the expected value of investing in the baseball game? 
 	Rain	No Rain	 
Probability	0.2	 	 
Leave money in bank	10000	10000
	 
Invest in  Baseball game 	0	23567

GPT-4o mini · Answer

To calculate the expected value of investing in the baseball game, we can use the formula for expected value, which is:

\[ \text{Expected Value (EV)} = (P(\text{No Rain}) \times \text{Payoff(No Rain)}) + (P(\text{Rain}) \times \text{Payoff(Rain)}) \]

Where:

$ P(\text{No Rain}) = 1 - P(\text{Rain}) = 1 - 0.2 = 0.8 $
Payoff(No Rain) = $23,567 (the revenue from selling tickets if the game goes ahead)
Payoff(Rain) = -$10,000 (the loss from the investment if it rains)

Now we can compute the expected value:

First, calculate each part of the expected value:

\[ \text{EV(No Rain)} = 0.8 \times 23567 = 18853.60 \]

\[ \text{EV(Rain)} = 0.2 \times (-10000) = -2000 \]

Now add these two results to find the total expected value:

\[ \text{EV} = 18853.60 - 2000 = 16853.60 \]

Thus, the expected value of investing in the baseball game is $16,853.60.

Since we also need to consider the initial investment of $10,000 in the bank as alternative:

The total value of leaving the money in the bank is $10,000$, while the expected value of investing is 16853.60 (profit over the investment).

Therefore, investing in the baseball game has a positive expected outcome compared to leaving the money in the bank, resulting in a better financial outcome in expectation.