To solve this problem, we calculate the expected value for the second bet and compare it to the first bet.

Part a: Calculating the Expected Value

The expected value ($EV$) is calculated using the formula: $EV = \sum (\text{Probability of Outcome} \times \text{Net Profit/Loss})$

For the $\$10$ bet on the outcome being $00$, $0$, or $1$:

• Probability of winning: $\frac{3}{38}$
• Probability of losing: $\frac{35}{38}$
• Net profit if winning: $+\$40$
• Loss if losing: $-\$10$

$EV = \left( \frac{3}{38} \times 40 \right) + \left( \frac{35}{38} \times (-10) \right)$

1. Compute each term:

   • $\frac{3}{38} \times 40 = \frac{120}{38} \approx 3.16$
   • $\frac{35}{38} \times (-10) = \frac{-350}{38} \approx -9.21$

2. Add the results: $EV \approx 3.16 - 9.21 = -6.05$

So, the expected value for the bet on $00$, $0$, or $1$ is approximately: $\boxed{-\$6.05}$

Part b: Comparing Bets

For the single-number bet ($29$), the expected value is already given as: $-\$0.53$

The expected value for the $00$, $0$, or $1$ bet is: $-\$6.05$

Since $-\$0.53$ is greater than $-\$6.05$, the bet on a single number (29) is better.

Final Answers

a. The expected value for the $\$10$ bet on $00$, $0$, or $1$ is: $\boxed{-\$6.05}$

b. The better bet is: $\boxed{\text{The single-number bet (29)}}$

Reason: The expected value of the single-number bet ($-\$0.53$) is closer to zero than the expected value of the other bet ($-\$6.05$), meaning it has a smaller average loss.

Do you want further clarification or calculations?

Five Related Questions:

1. How is expected value used in determining the fairness of a game?
2. What does a negative expected value imply for gambling outcomes?
3. How would the probabilities and expected values change if the casino used a European roulette wheel instead?
4. Can expected value alone determine whether a bet is worth making?
5. How does increasing the bet amount affect the expected value in this scenario?

Tip:

When gambling, always compare the expected values of bets to minimize losses and make more informed decisions.