Jim loves to play roulette! He bets $10 on a single number, usually the number 13. He has a 138 chance of winning. If the ball lands on his number, he wins $350 (and gets to keep his $10 bet!). If it lands on any other number, a 3738 probability, he loses his bet. What is the expected value of a game of roulette?

Round to the nearest cent. Do not round until your final answer. 

To calculate the expected value (EV) of Jim's roulette game, we can break it down using the given probabilities and outcomes.

1. **Winning Scenario**:
   - Probability of winning (landing on 13): $ P(\text{Win}) = \frac{1}{38} $
   - Winnings: $350 (plus his $10 bet back, for a total of $360)
   - Expected value from winning:  
   \[
   EV_{\text{win}} = P(\text{Win}) \times \text{Winnings} = \frac{1}{38} \times 360
   \]

2. **Losing Scenario**:
   - Probability of losing (landing on any other number): $ P(\text{Lose}) = \frac{37}{38} $
   - Loss: $10
   - Expected value from losing:  
   \[
   EV_{\text{lose}} = P(\text{Lose}) \times \text{Loss} = \frac{37}{38} \times (-10)
   \]

3. **Total Expected Value**:
   - Combine both scenarios:
   \[
   EV = EV_{\text{win}} + EV_{\text{lose}}
   \]
   \[
   EV = \left(\frac{1}{38} \times 360\right) + \left(\frac{37}{38} \times (-10)\right)
   \]

Let's calculate this step by step.

### Step 1: Calculate $ EV_{\text{win}} $
\[
EV_{\text{win}} = \frac{1}{38} \times 360 = \frac{360}{38} \approx 9.47368421
\]

### Step 2: Calculate $ EV_{\text{lose}} $
\[
EV_{\text{lose}} = \frac{37}{38} \times (-10) = \frac{-370}{38} \approx -9.73684211
\]

### Step 3: Calculate Total Expected Value
\[
EV = 9.47368421 - 9.73684211 \approx -0.26315790
\]

### Final Result
Rounding to the nearest cent:
\[
\text{Expected Value} \approx -0.26
\]

Thus, the expected value of a game of roulette for Jim is approximately **-$0.26**.

Would you like more details or have any questions? Here are some related questions you might consider:

1. What factors can influence the outcome in a game of roulette?
2. How does the house edge affect expected value in casino games?
3. What is the difference between American and European roulette?
4. How can understanding expected value help in gambling decisions?
5. What are some strategies players use in roulette? 

**Tip**: Always understand the probabilities involved before placing bets in games of chance!

Jim loves to play roulette! He bets $10 on a single number, usually the number 13. He has a 138 chance of winning. If the ball lands on his number, he wins $350 (and gets to keep his $10 bet!). If it lands on any other number, a 3738 probability, he loses his bet. What is the expected value of a game of roulette?

Discover how to calculate the expected value of a roulette game when betting on number 13. Jim's expected value analysis shows a winning probability of 1/38 and a losing probability of 37/38, leading to an expected loss of approximately -$0.26.

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