In the game of​ roulette, a player can place a

​$99

bet on the number

99

and have a

StartFraction 1 Over 38 EndFraction138

probability of winning. If the metal ball lands on

99​,

the player gets to keep the

​$99

paid to play the game and the player is awarded an additional

​$315315.

​ Otherwise, the player is awarded nothing and the casino takes the​ player's

​$99.

What is the expected value of the game to the​ player? If you played the game 1000​ times, how much would you expect to​ lose?

Question content area bottom

Part 1

The expected value is

​$enter your response here.

### 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**.

---

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!

In the game of roulette, a player can place a $99 bet on the number 99 and have a 1/38 probability of winning. If the metal ball lands on 99, the player gets to keep the $99 paid to play the game and the player is awarded an additional $315. Otherwise, the player is awarded nothing and the casino takes the player's $99. What is the expected value of the game to the player? If you played the game 1000 times, how much would you expect to lose?

Understand how to calculate the expected value of a $99 bet on the number 99 in roulette. Using probability and expected value formulas, this detailed solution walks through the step-by-step calculation of expected gains and losses in the game. Suitable for Grades 10-12 students learning probability and game theory.

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