Part 1: Finding the Expected Value

To find the expected value of the game, we need to consider both the probability of winning and the probability of losing.

Winning Scenario:

• The probability of winning is $\frac{1}{1000}$ because there are 1000 possible numbers, and the player wins only if their chosen number is selected.
• The payout for winning is $2300.

Losing Scenario:

• The probability of losing is $\frac{999}{1000}$ because there are 999 other possible numbers.
• The player loses their $5 bet in this case.

Expected Value Formula:

$\text{Expected Value (EV)} = ( \text{Probability of Winning} \times \text{Payout for Winning} ) + ( \text{Probability of Losing} \times \text{Loss for Losing} )$

$EV = \left( \frac{1}{1000} \times 2300 \right) + \left( \frac{999}{1000} \times (-5) \right)$

Calculating each part:

1. The expected winnings for winning: $\frac{1}{1000} \times 2300 = 2.30$

2. The expected loss for losing: $\frac{999}{1000} \times (-5) = -4.995$

Now, summing these values:

$EV = 2.30 + (-4.995) = -2.695$

Rounding to the nearest cent:

$EV = -2.70$

Part 2: Interpretation of the Expected Value

The expected value of $-2.70$ means that, on average, the player loses $2.70 for each game played in the long run. Therefore, the correct interpretation is:

A. In the long​ run, the player can expect to lose $2.70 for each game played.

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Summary

1. The expected value for this game is $-2.70.
2. The best interpretation is A. In the long​ run, the player can expect to lose $2.70 for each game played.

Would you like more details on how this formula is derived or how probabilities work in similar games?

Related Questions:

1. How can you calculate the expected value for a different type of gambling game?
2. What are the key factors that influence the expected value in a game of chance?
3. How does increasing the payout affect the expected value?
4. What is the significance of a negative expected value in gambling games?
5. Can you calculate the expected value for a similar game with more players or different probabilities?

Tip: When dealing with expected value, remember that it is a long-term average, so it might not represent the outcome of one single game but rather many games over time.