Let's break down the problem to determine the expected value for both the pennies and the nickels based on the game's rules.
### Analyzing the Pennies:
- All pennies heads: The probability of getting all heads with 3 pennies is [tex]\(\left(\frac{1}{2}\right)^3 = \frac{1}{8}\)[/tex]. The points earned is -2.
- At least one of each: This means having a mix of heads and tails. The probability of this outcome is calculated by subtracting the probabilities of all heads and all tails from 1:
[tex]\[
1 - \left(\frac{1}{8} + \frac{1}{8}\right) = \frac{6}{8} = \frac{3}{4}
\][/tex]
The points earned is +3.
- All pennies tails: The probability of getting all tails with 3 pennies is [tex]\(\left(\frac{1}{2}\right)^3 = \frac{1}{8}\)[/tex]. The points earned is -2.
Now, we can calculate the expected value [tex]\(E\)[/tex] for the pennies:
[tex]\[
E_{penny} = (-2) \times \frac{1}{8} + 3 \times \frac{3}{4} - 2 \times \frac{1}{8 }
\][/tex]
### Analyzing the Nickels:
- Both nickels heads: The probability of getting both heads with 2 nickels is [tex]\(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\)[/tex]. The points earned is -2.
- One of each: This means having one head and one tail. The probability of this outcome is:
[tex]\[
\left(\frac{1}{2} \times \frac{1}{2}\right) + \left(\frac{1}{2} \times \frac{1}{2}\right) = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}
\][/tex]
The points earned is +5.
- Both nickels tails: The probability of getting both tails with 2 nickels is [tex]\(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\)[/tex]. The points earned is -2.
Now, we can calculate the expected value [tex]\(E\)[/tex] for the nickels:
[tex]\[
E_{nickel} = (-2) \times \frac{1}{4} + 5 \times \frac{1}{2} - 2 \times \frac{1}{4}
\][/tex]
Given the calculations:
- The expected value [tex]\(E\)[/tex] for the pennies is [tex]\(1.75\)[/tex].
- The expected value [tex]\(E\)[/tex] for the nickels is [tex]\(1.5\)[/tex].
Comparing the two expected values:
[tex]\[
E(penny) = 1.75 \quad \text{and} \quad E(nickel) = 1.5
\][/tex]
Based on these values, Alyssa should choose to play with the pennies because the expected value of playing with pennies is higher.
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]
