To solve this, we'll break the problem down into two parts:

Part 1: Expected Value Calculation

The expected value $E$ for a game of chance can be calculated using the following formula:

$E = (P_{\text{win}} \times \text{win amount}) + (P_{\text{lose}} \times \text{lose amount})$

Where:

• $P_{\text{win}}$ is the probability of winning.
• $\text{win amount}$ is the net amount won, including the player's original bet.
• $P_{\text{lose}}$ is the probability of losing.
• $\text{lose amount}$ is the amount lost when the player loses.

Step 1: Define the values.

• The probability of winning is $P_{\text{win}} = \frac{1}{38}$.
• The probability of losing is $P_{\text{lose}} = 1 - P_{\text{win}} = \frac{37}{38}$.
• The win amount is $140 + 44 = 184$ (since the player gets their initial bet of $44 back plus $140 in winnings).
• The lose amount is $-44$ (the player loses their bet).

Step 2: Apply the values to the formula.

$E = \left( \frac{1}{38} \times 184 \right) + \left( \frac{37}{38} \times (-44) \right)$

$E = \left( \frac{184}{38} \right) + \left( \frac{-1628}{38} \right)$

$E = 4.84 + (-42.84)$

$E = -38$

So, the expected value for the game is -38 dollars. This means that on average, the player expects to lose $38 per game.

Part 2: Expected Loss Over 1000 Games

To find the expected loss over 1000 games, simply multiply the expected value per game by 1000:

$\text{Expected loss over 1000 games} = -38 \times 1000 = -38000$

So, if you played 1000 games, you would expect to lose about $38,000.

Final Answers:

1. The expected value of the game is -38 dollars.
2. The player would expect to lose about $38,000 if they played 1000 times.

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Would you like more details or have any questions about this solution?

Here are 5 related questions to consider:

1. How does increasing the number of bets affect the expected value in gambling games?
2. What would happen to the expected value if the casino paid out $200 instead of $140 for a win?
3. Can a player ever expect to win money in a game with a negative expected value?
4. How is expected value used in decision-making in other areas like finance?
5. If the probability of winning changes, how does it affect the overall expected loss?

Tip: Understanding expected value helps in evaluating risk in games, investments, and everyday decisions.