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!