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Let's address each part of the question step-by-step:
### Part (a) - Finding the Expected Value for the [tex]$20 Bet on 00, 0, or 1 The expected value is calculated by considering all possible outcomes, their probabilities, and the respective payoffs. 1. Determine probabilities and outcomes: - Probability of winning (betting on 00, 0, or 1): \( \frac{3}{38} \) - Probability of losing (betting on anything else): \( \frac{35}{38} \) 2. Determine payoffs: - Net profit if the bet wins: \( +\$[/tex]60 \)
- Net loss if the bet loses: [tex]\( -\$20 \)[/tex]
3. Calculate the expected value:
The expected value (EV) is a weighted average of all possible outcomes, expressed as:
[tex]\[ \text{EV} = \left(\text{Probability of winning} \times \text{Net profit}\right) + \left(\text{Probability of losing} \times \text{Net loss}\right) \][/tex]
Plug in the respective values:
[tex]\[ \text{EV} = \left(\frac{3}{38} \times 60\right) + \left(\frac{35}{38} \times -20\right) \][/tex]
Simplify this expression:
[tex]\[ \text{EV} = \left(\frac{3 \times 60}{38}\right) + \left(\frac{35 \times -20}{38}\right) \][/tex]
[tex]\[ \text{EV} = \left(\frac{180}{38}\right) + \left(\frac{-700}{38}\right) \][/tex]
[tex]\[ \text{EV} = \frac{180 - 700}{38} \][/tex]
[tex]\[ \text{EV} = \frac{-520}{38} \][/tex]
[tex]\[ \text{EV} \approx -13.684210526315788 \][/tex]
Thus, the expected value for the \[tex]$20 bet on 00, 0, or 1 is approximately \(-\$[/tex]13.68421\).
### Part (b) - Comparing Bets: [tex]$20 Bet on Number 11 vs. $[/tex]20 Bet on 00, 0, or 1
You are given that the expected value of the \[tex]$20 bet on the number 11 is \(-\$[/tex]1.05\).
1. Expected value for the bet on 00, 0, or 1: [tex]\( -\$13.68421 \)[/tex]
2. Expected value for the bet on number 11: [tex]\( -\$1.05 \)[/tex]
### Conclusion
Compare the expected values:
- Expected value for betting on 00, 0, or 1: [tex]\( -\$13.68421 \)[/tex]
- Expected value for betting on number 11: [tex]\( -\$1.05 \)[/tex]
Since [tex]\(-\$1.05\)[/tex] is greater (less negative) than [tex]\(-\$13.68421\)[/tex], the expected loss is smaller when betting on the number 11.
Thus, the \$20 bet on the number 11 is the better bet because it has a higher expected value, meaning a smaller expected loss compared to betting on 00, 0, or 1.
### Part (a) - Finding the Expected Value for the [tex]$20 Bet on 00, 0, or 1 The expected value is calculated by considering all possible outcomes, their probabilities, and the respective payoffs. 1. Determine probabilities and outcomes: - Probability of winning (betting on 00, 0, or 1): \( \frac{3}{38} \) - Probability of losing (betting on anything else): \( \frac{35}{38} \) 2. Determine payoffs: - Net profit if the bet wins: \( +\$[/tex]60 \)
- Net loss if the bet loses: [tex]\( -\$20 \)[/tex]
3. Calculate the expected value:
The expected value (EV) is a weighted average of all possible outcomes, expressed as:
[tex]\[ \text{EV} = \left(\text{Probability of winning} \times \text{Net profit}\right) + \left(\text{Probability of losing} \times \text{Net loss}\right) \][/tex]
Plug in the respective values:
[tex]\[ \text{EV} = \left(\frac{3}{38} \times 60\right) + \left(\frac{35}{38} \times -20\right) \][/tex]
Simplify this expression:
[tex]\[ \text{EV} = \left(\frac{3 \times 60}{38}\right) + \left(\frac{35 \times -20}{38}\right) \][/tex]
[tex]\[ \text{EV} = \left(\frac{180}{38}\right) + \left(\frac{-700}{38}\right) \][/tex]
[tex]\[ \text{EV} = \frac{180 - 700}{38} \][/tex]
[tex]\[ \text{EV} = \frac{-520}{38} \][/tex]
[tex]\[ \text{EV} \approx -13.684210526315788 \][/tex]
Thus, the expected value for the \[tex]$20 bet on 00, 0, or 1 is approximately \(-\$[/tex]13.68421\).
### Part (b) - Comparing Bets: [tex]$20 Bet on Number 11 vs. $[/tex]20 Bet on 00, 0, or 1
You are given that the expected value of the \[tex]$20 bet on the number 11 is \(-\$[/tex]1.05\).
1. Expected value for the bet on 00, 0, or 1: [tex]\( -\$13.68421 \)[/tex]
2. Expected value for the bet on number 11: [tex]\( -\$1.05 \)[/tex]
### Conclusion
Compare the expected values:
- Expected value for betting on 00, 0, or 1: [tex]\( -\$13.68421 \)[/tex]
- Expected value for betting on number 11: [tex]\( -\$1.05 \)[/tex]
Since [tex]\(-\$1.05\)[/tex] is greater (less negative) than [tex]\(-\$13.68421\)[/tex], the expected loss is smaller when betting on the number 11.
Thus, the \$20 bet on the number 11 is the better bet because it has a higher expected value, meaning a smaller expected loss compared to betting on 00, 0, or 1.
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