Part A: Finding the Expected Value for the Bet on 00, 0, 1, 2, or 3

The expected value (EV) of a bet is calculated using the formula:

$EV = ( \text{probability of winning} \times \text{net gain from winning} ) + ( \text{probability of losing} \times \text{net loss from losing} )$

For the bet on the five numbers $00, 0, 1, 2, 3$:

• Probability of winning: $\frac{5}{38}$
• Net gain from winning: $30 (since the gambler gains $30 for a win)
• Probability of losing: $\frac{33}{38}$
• Net loss from losing: $55

Now, plug these values into the formula:

$EV = \left( \frac{5}{38} \times 30 \right) + \left( \frac{33}{38} \times (-55) \right)$

Calculate each part:

1. $\frac{5}{38} \times 30 = \frac{150}{38} \approx 3.95$
2. $\frac{33}{38} \times (-55) = \frac{-1815}{38} \approx -47.76$

So, the expected value is:

$EV = 3.95 + (-47.76) = -43.81$

Thus, the expected value for the bet on the numbers $00, 0, 1, 2, 3$ is approximately -43.81 dollars.

Part B: Which Bet is Better?

Now, let's compare the expected values of the two bets:

• The expected value of the $55 bet on number 32 is -53 cents or -0.53 dollars.
• The expected value of the $55 bet on numbers 00, 0, 1, 2, 3 is -43.81 dollars.

Conclusion: The bet on the number 32 is a better bet because its expected loss is only 53 cents, whereas the expected loss for the bet on the five numbers is much larger at 43.81 dollars.

The gambler loses less money on average by betting on a single number (32) compared to betting on multiple numbers (00, 0, 1, 2, 3).

Would you like further details or have any questions?

Here are 5 related questions to think about:

1. How is the expected value different if the bet amount is changed to $100?
2. How would the expected value change if the number of possible outcomes in roulette changed?
3. What would happen to the expected value if the payout for winning on the 5 numbers was higher than $30?
4. How is expected value used in other gambling games like blackjack or poker?
5. What is the long-term financial impact of repeatedly making bets with negative expected values?

Tip: Always assess the expected value before placing a bet. It helps you understand your average outcome over time, especially for high-risk games like roulette.