When playing roulette at a​ casino, a gambler is trying to decide whether to bet

​$55

on the number

3232

or to bet

​$55

that the outcome is any one of the

fivefive

possibilities

00 comma 0 comma 1 comma 2 comma or 300, 0, 1, 2, or 3.

The gambler knows that the expected value of the

​$55

bet for a single number is

negative 53 cents−53¢.

For the

​$55

bet that the outcome is

00 comma 0 comma 1 comma 2 comma or 300, 0, 1, 2, or 3​,

there is a probability of

StartFraction 5 Over 38 EndFraction538

of making a net profit of

​$3030

and a

StartFraction 33 Over 38 EndFraction3338

probability of losing

​$55.

**a.** Find the expected value for the

​$55

bet that the outcome is

00 comma 0 comma 1 comma 2 comma or 300, 0, 1, 2, or 3.

**b.** Which bet is​ better: a

​$55

bet on the number

3232

or a

​$55

bet that the outcome is any one of the numbers

00 comma 0 comma 1 comma 2 comma or 300, 0, 1, 2, or 3​?

​Why?

### 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.

When playing roulette at a casino, a gambler is trying to decide whether to bet $55 on the number 32 or to bet $55 that the outcome is any one of the five possibilities 00, 0, 1, 2, or 3. The gambler knows that the expected value of the $55 bet for a single number is -53¢. For the $55 bet that the outcome is 00, 0, 1, 2, or 3, there is a probability of 5/38 of making a net profit of $30 and a probability of 33/38 of losing $55.

 a. Find the expected value for the $55 bet that the outcome is 00, 0, 1, 2, or 3.

 b. Which bet is better: a $55 bet on the number 32 or a $55 bet that the outcome is any one of the numbers 00, 0, 1, 2, or 3? Why?

Discover how to calculate the expected value of two roulette bets: a $55 bet on the number 32 and a $55 bet on 00, 0, 1, 2, or 3. This solution explains key probability concepts and uses the expected value formula to determine which bet minimizes losses.

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