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Let's analyze the given game step by step to find out the expected values for playing with the 3 pennies and the 2 nickels.
### Step 1: Calculate the Probabilities
#### Probabilities for Pennies:
1. All pennies heads:
- Probability: [tex]\((\frac{1}{2})^3 = \frac{1}{8}\)[/tex].
2. All pennies tails:
- Probability: [tex]\((\frac{1}{2})^3 = \frac{1}{8}\)[/tex].
3. At least one of each (mixed):
- Probability: [tex]\(1 - \left( \frac{1}{8} + \frac{1}{8} \right) = 1 - \frac{2}{8} = \frac{6}{8} = \frac{3}{4}\)[/tex].
#### Probabilities for Nickels:
1. Both nickels heads:
- Probability: [tex]\((\frac{1}{2})^2 = \frac{1}{4}\)[/tex].
2. Both nickels tails:
- Probability: [tex]\((\frac{1}{2})^2 = \frac{1}{4}\)[/tex].
3. One of each (mixed):
- Probability: [tex]\(1 - \left( \frac{1}{4} + \frac{1}{4} \right) = 1 - \frac{2}{4} = \frac{2}{4} = \frac{1}{2}\)[/tex].
### Step 2: Calculate the Expected Points
#### Expected Points for Pennies:
1. All pennies heads: [tex]\(-2\)[/tex] points.
- Contribution: [tex]\(\frac{1}{8} \times -2 = -\frac{2}{8} = -0.25\)[/tex].
2. All pennies tails: [tex]\(-2\)[/tex] points.
- Contribution: [tex]\(\frac{1}{8} \times -2 = -\frac{2}{8} = -0.25\)[/tex].
3. Mixed: [tex]\(+3\)[/tex] points.
- Contribution: [tex]\(\frac{3}{4} \times 3 = \frac{9}{4} = 2.25\)[/tex].
Adding these contributions:
[tex]\[ E(\text{penny}) = -0.25 - 0.25 + 2.25 = 1.75 \][/tex]
#### Expected Points for Nickels:
1. Both nickels heads: [tex]\(-2\)[/tex] points.
- Contribution: [tex]\(\frac{1}{4} \times -2 = -\frac{2}{4} = -0.5\)[/tex].
2. Both nickels tails: [tex]\(-2\)[/tex] points.
- Contribution: [tex]\(\frac{1}{4} \times -2 = -\frac{2}{4} = -0.5\)[/tex].
3. Mixed: [tex]\(+5\)[/tex] points.
- Contribution: [tex]\(\frac{1}{2} \times 5 = \frac{5}{2} = 2.5\)[/tex].
Adding these contributions:
[tex]\[ E(\text{nickel}) = -0.5 - 0.5 + 2.5 = 1.5 \][/tex]
### Step 3: Compare Expected Values and Make a Decision
The expected value for playing with pennies is [tex]\(1.75\)[/tex], and for playing with nickels is [tex]\(1.5\)[/tex].
Since [tex]\(1.75 > 1.5\)[/tex], Alyssa should play with the pennies.
### Conclusion
The correct statement is:
[tex]\[ E(\text{penny}) = 1.75 \text{ and } E(\text{nickel}) = 1.5, \text{ so she should play with the pennies.} \][/tex]
### Step 1: Calculate the Probabilities
#### Probabilities for Pennies:
1. All pennies heads:
- Probability: [tex]\((\frac{1}{2})^3 = \frac{1}{8}\)[/tex].
2. All pennies tails:
- Probability: [tex]\((\frac{1}{2})^3 = \frac{1}{8}\)[/tex].
3. At least one of each (mixed):
- Probability: [tex]\(1 - \left( \frac{1}{8} + \frac{1}{8} \right) = 1 - \frac{2}{8} = \frac{6}{8} = \frac{3}{4}\)[/tex].
#### Probabilities for Nickels:
1. Both nickels heads:
- Probability: [tex]\((\frac{1}{2})^2 = \frac{1}{4}\)[/tex].
2. Both nickels tails:
- Probability: [tex]\((\frac{1}{2})^2 = \frac{1}{4}\)[/tex].
3. One of each (mixed):
- Probability: [tex]\(1 - \left( \frac{1}{4} + \frac{1}{4} \right) = 1 - \frac{2}{4} = \frac{2}{4} = \frac{1}{2}\)[/tex].
### Step 2: Calculate the Expected Points
#### Expected Points for Pennies:
1. All pennies heads: [tex]\(-2\)[/tex] points.
- Contribution: [tex]\(\frac{1}{8} \times -2 = -\frac{2}{8} = -0.25\)[/tex].
2. All pennies tails: [tex]\(-2\)[/tex] points.
- Contribution: [tex]\(\frac{1}{8} \times -2 = -\frac{2}{8} = -0.25\)[/tex].
3. Mixed: [tex]\(+3\)[/tex] points.
- Contribution: [tex]\(\frac{3}{4} \times 3 = \frac{9}{4} = 2.25\)[/tex].
Adding these contributions:
[tex]\[ E(\text{penny}) = -0.25 - 0.25 + 2.25 = 1.75 \][/tex]
#### Expected Points for Nickels:
1. Both nickels heads: [tex]\(-2\)[/tex] points.
- Contribution: [tex]\(\frac{1}{4} \times -2 = -\frac{2}{4} = -0.5\)[/tex].
2. Both nickels tails: [tex]\(-2\)[/tex] points.
- Contribution: [tex]\(\frac{1}{4} \times -2 = -\frac{2}{4} = -0.5\)[/tex].
3. Mixed: [tex]\(+5\)[/tex] points.
- Contribution: [tex]\(\frac{1}{2} \times 5 = \frac{5}{2} = 2.5\)[/tex].
Adding these contributions:
[tex]\[ E(\text{nickel}) = -0.5 - 0.5 + 2.5 = 1.5 \][/tex]
### Step 3: Compare Expected Values and Make a Decision
The expected value for playing with pennies is [tex]\(1.75\)[/tex], and for playing with nickels is [tex]\(1.5\)[/tex].
Since [tex]\(1.75 > 1.5\)[/tex], Alyssa should play with the pennies.
### Conclusion
The correct statement is:
[tex]\[ E(\text{penny}) = 1.75 \text{ and } E(\text{nickel}) = 1.5, \text{ so she should play with the pennies.} \][/tex]
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