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Two students devised a game called "3 Pennies & 2 Nickels." Each player will choose to play with the pennies or the nickels. In each round, the players will flip all their coins on the table and record how many heads and tails they have. The table below includes the point scheme.

\begin{tabular}{|l|l|}
\hline \multicolumn{2}{|c|}{ Point Values for "3 Pennies \& 2 Nickels" } \\
\hline Penny Points & Nickel Points \\
\hline All pennies heads: -2 & Both nickels heads: -2 \\
\hline At least one of each: +3 & One of each: +5 \\
\hline All pennies tails: -2 & Both nickels tails: -2 \\
\hline \hline
\end{tabular}

Alyssa wants to play the game. Choose the statement below that is correct in all aspects.

A. [tex]E(\text{penny}) = -0.33[/tex] and [tex]E(\text{nickel}) = 0.33[/tex], so she should play with the nickels.
B. [tex]E(\text{penny}) = 0.5[/tex] and [tex]E(\text{nickel}) = 1.5[/tex], so she should play with the nickels.
C. [tex]E(\text{penny}) = 1.75[/tex] and [tex]E(\text{nickel}) = 1.5[/tex], so she should play with the pennies.
D. [tex]E(\text{penny}) = 2.38[/tex] and [tex]E(\text{nickel}) = 2[/tex], so she should play with the pennies.



Answer :

GinnyAnswer
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]