In roulette, a three-number bet (also known as a "street" bet) involves betting on three numbers in a row on the roulette table. In American roulette, there are 38 possible outcomes (numbers 1-36, 0, and 00), and for a three-number bet, if the bet wins, it pays 11 to 1.

Let's solve each part of the problem:

(a) Expected Value of Each $1 Bet

1. Probability of Winning: The probability of winning with a three-number bet is: $\frac{\text{number of winning outcomes}}{\text{total outcomes}} = \frac{3}{38}$

2. Probability of Losing: The probability of losing the bet is: $1 - \frac{3}{38} = \frac{35}{38}$

3. Payout if Win: If you win, the casino pays 11 to 1, so for a $1 bet, the winnings (including the original bet) would be $12.

4. Expected Value Calculation: Expected value $E$ is calculated as follows: $E = (\text{win amount} \times \text{probability of winning}) + (\text{loss amount} \times \text{probability of losing})$ Substituting values: $E = (12 \times \frac{3}{38}) + (-1 \times \frac{35}{38})$

   Let's calculate this:

   $E = \left(\frac{36}{38}\right) - \left(\frac{35}{38}\right) = \frac{1}{38} \approx 0.026$

So, the expected value of each $1 bet is approximately $-0.053 (rounded to three decimal places).

(b) Interpretation

Over time, you should expect to lose about 5.3 cents for every dollar you bet.

Let me know if you would like further clarification on any part of the calculation!

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Further Questions:

1. How would the expected value change with a European roulette wheel with only one "0"?
2. What is the expected value for other types of bets in roulette, like single-number bets?
3. How does the house edge in roulette compare to other casino games?
4. How would increasing the payout for a three-number bet affect the expected value?
5. If you bet multiple times, how does the law of large numbers affect your actual losses?

Tip:

In games of chance, understanding expected value helps gauge the long-term profit or loss in a given bet structure.