Part 1: Calculate the Expected Value (EV)

The expected value (EV) of a game can be calculated as follows:

$\text{EV} = P(\text{win}) \times \text{Winnings} + P(\text{lose}) \times \text{Losses}$

In this case:

• Probability of winning ($P(\text{win})$): $\frac{1}{38}$
• Probability of losing ($P(\text{lose})$): $\frac{37}{38}$ (since there are 38 total outcomes and only 1 is favorable)

Winnings:

• The player keeps the $99 and receives an additional $315, so the total winnings are: $\text{Winnings} = 99 + 315 = 414$

Losses:

• If the player loses, they lose the $99 bet: $\text{Losses} = -99$

Step-by-step Calculation

Expected value of winning:

$EV(\text{win}) = \frac{1}{38} \times 414 = \frac{414}{38} = 10.89$

Expected value of losing:

$EV(\text{lose}) = \frac{37}{38} \times (-99) = \frac{-3663}{38} = -96.39$

Total expected value:

$EV(\text{total}) = 10.89 - 96.39 = -85.50$

Thus, the expected value of the game to the player is $\mathbf{-85.50}$.

Part 2: Expected Loss Over 1000 Games

If you play the game 1000 times, the total expected loss would be: $1000 \times -85.50 = \mathbf{-85,500}$

Therefore, if you played 1000 times, you would expect to lose $85,500.

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

5 Related Questions:

1. How is expected value used in decision-making for gambling?
2. What are the probabilities of winning in other roulette bets?
3. How do casinos use expected value to ensure profit?
4. Can the expected value ever be positive in roulette games?
5. How do the odds in European roulette compare to American roulette?

Tip:

When gambling, always consider the expected value to understand long-term outcomes instead of short-term wins or losses!