When playing roulette at a​ casino, a gambler is trying to decide whether to bet ​$15 on the number 31 or to bet ​$15 that the outcome is any one of the three possibilities 00 comma 0 comma or 1. The gambler knows that the expected value of the ​$15 bet for a single number is negative $ 1.58. For the ​$15 bet that the outcome is 00 comma 0 comma or 1​, there is a probability of StartFraction 3 Over 38 EndFraction
of making a net profit of ​$45 and a StartFraction 35 Over 38 EndFraction
probability of losing ​$15.
a. Find the expected value for the ​$15 bet that the outcome is 00 comma 0 comma or 1.
b. Which bet is​ better: a ​$15 bet on the number 31 or a ​$15 bet that the outcome is any one of the numbers 00 comma 0 comma or 1​? ​Why?
Question content area bottom
Part 1
a. The expected value is ​$

enter your response here.
​(Round to the nearest cent as​ needed.)

Let's work through this step by step.

### Part 1: a. Finding the expected value for the $ \$15 $ bet that the outcome is 00, 0, or 1.

The expected value (EV) of a bet is given by the formula:

\[
EV = (P(\text{win}) \times \text{profit}) + (P(\text{loss}) \times \text{loss})
\]

Where:
- $ P(\text{win}) = \frac{3}{38} $ (probability of hitting 00, 0, or 1),
- $ P(\text{loss}) = \frac{35}{38} $ (probability of not hitting 00, 0, or 1),
- The profit on a win is $ \$45 $,
- The loss is $ -\$15 $.

Now, let's calculate the expected value.

1. **Probability of winning**: $ P(\text{win}) = \frac{3}{38} $
   \[
   \frac{3}{38} \times 45 = \frac{135}{38} = 3.55
   \]

2. **Probability of losing**: $ P(\text{loss}) = \frac{35}{38} $
   \[
   \frac{35}{38} \times (-15) = \frac{-525}{38} = -13.82
   \]

3. **Expected Value**:
   \[
   EV = 3.55 + (-13.82) = -10.27
   \]

Therefore, the expected value for the $ \$15 $ bet that the outcome is 00, 0, or 1 is **\$-10.27**.

---

### Part 2: b. Which bet is better?

- The expected value for a $ \$15 $ bet on number 31 is $ -1.58 $.
- The expected value for the $ \$15 $ bet on 00, 0, or 1 is $ -10.27 $.

Since the expected value of betting on the number 31 is **closer to zero** (i.e., less negative) than betting on 00, 0, or 1, **the $ \$15 $ bet on the number 31 is better**. This is because a less negative expected value means that, on average, the losses are smaller.

### Conclusion:
- **a.** The expected value for the $ \$15 $ bet on 00, 0, or 1 is **\$-10.27**.
- **b.** The $ \$15 $ bet on number 31 is better because it has a higher expected value (\$-1.58 vs. \$-10.27).

Would you like any further details on how this was calculated? 

Here are 5 related questions:
1. How is expected value applied in real-world casino strategies?
2. What is the house edge in roulette and how does it affect expected value?
3. How does the number of possible outcomes in roulette influence bet choices?
4. Can expected value be used to predict long-term gambling outcomes?
5. What are examples of games with positive expected values for players?

Tip: Understanding expected value can help you make smarter financial decisions, not just in gambling, but also in investing and business choices.

When playing roulette at a casino, a gambler is trying to decide whether to bet $15 on the number 31 or to bet $15 that the outcome is any one of the three possibilities 00, 0, or 1. The gambler knows that the expected value of the $15 bet for a single number is -$1.58. For the $15 bet that the outcome is 00, 0, or 1, there is a probability of 3/38 of making a net profit of $45 and a 35/38 probability of losing $15.

a. Find the expected value for the $15 bet that the outcome is 00, 0, or 1.
b. Which bet is better: a $15 bet on the number 31 or a $15 bet that the outcome is any one of the numbers 00, 0, or 1? Why?

Discover how to calculate the expected value of a $15 bet on either the numbers 00, 0, or 1, compared to betting on the number 31 in roulette. The problem explores probability and expected value concepts, with the expected value of the $15 bet on 00, 0, or 1 being -$10.27, and the expected value of betting on 31 being -$1.58. Ideal for students in Grades 11-12.

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