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When playing roulette at a casino, a gambler is trying to decide whether to bet [tex]\$20[/tex] on the number 36 or to bet [tex]\$20[/tex] that the outcome is any one of the three possibilities 00, 0, or 1. The gambler knows that the expected value of the [tex]\$20[/tex] bet for a single number is [tex]-\$2.10[/tex]. For the [tex]\$20[/tex] bet that the outcome is 00, 0, or 1, there is a probability of [tex]\frac{3}{38}[/tex] of making a net profit of [tex]\$80[/tex] and a [tex]\frac{35}{38}[/tex] probability of losing [tex]\[tex]$20[/tex].

a. Find the expected value for the [tex]\$[/tex]20[/tex] bet that the outcome is 00, 0, or 1.
b. Which bet is better: a [tex]\$20[/tex] bet on the number 36 or a [tex]\$20[/tex] bet that the outcome is any one of the numbers 00, 0, or 1? Why?

a. The expected value is [tex]\$\square[/tex] (Round to the nearest cent as needed.)

b. Since the expected value of the bet on the number 36 is [tex]\square[/tex] than the expected value for the bet that the outcome is 00, 0, or 1, the bet on [tex]\square[/tex] is better.



Answer :

GinnyAnswer
Sure, let's tackle this problem step by step:

### Part (a): Finding the Expected Value

1. Define the probabilities and outcomes:
- Probability of winning for 00, 0, or 1: [tex]\(\frac{3}{38}\)[/tex]
- Probability of losing: [tex]\(\frac{35}{38}\)[/tex]
- Net profit if the gambler wins: [tex]\( \$ 80 \)[/tex]
- Net loss if the gambler loses: [tex]\( -\$ 20 \)[/tex]

2. Calculate the Expected Value (EV):
The formula for expected value (EV) is:
[tex]\[ EV = ( \text{Probability of Winning} \times \text{Net Profit}) + ( \text{Probability of Losing} \times \text{Net Loss}) \][/tex]

3. Substitute the values:
[tex]\[ EV = \left(\frac{3}{38} \times 80 \right) + \left(\frac{35}{38} \times -20 \right) \][/tex]

4. Calculate the expected value step by step:
[tex]\[ EV = \left(\frac{3}{38} \times 80 \right) + \left(\frac{35}{38} \times -20 \right) \][/tex]
[tex]\[ EV = \left(\frac{240}{38} \right) + \left(\frac{-700}{38} \right) \][/tex]
[tex]\[ EV = \left(\frac{240 - 700}{38} \right) \][/tex]
[tex]\[ EV = \left(\frac{-460}{38} \right) \][/tex]
[tex]\[ EV \approx -12.11 \][/tex]
Thus, the expected value for the \[tex]$20 bet that the outcome is 00, 0, or 1 is \(-\$[/tex]12.11\).

### Part (b): Comparing the Bets

1. Given Information:
- Expected value for a [tex]$20 bet on the number 36 is \( -\$[/tex]2.10 \).

2. Determine which expected value is better:
- Expected value for 00, 0, or 1: [tex]\( -\$12.11 \)[/tex]
- Expected value for the number 36: [tex]\( -\$2.10 \)[/tex]

3. Comparison:
- [tex]\(-\$2.10\)[/tex] (EV for betting on 36) is greater than [tex]\(-\$12.11\)[/tex] (EV for betting on 00, 0, or 1).

4. Conclusion:
- The bet on the number 36 is a better choice because its expected value is higher (less negative) than the other option.

### Final Answer:
a. The expected value is [tex]\(-\$12.11\)[/tex].

b. Since the expected value of the bet on the number 36 is greater than the expected value for the bet that the outcome is 00, 0, or 1, the bet on 36 is better.

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