Question

g.pr.10.5 (mc)

a coin is tossed 3 times. you earn 2 points if there is only one head, 5 points for two heads, and 10 points for 3 heads. the probability for these possibilities is below. what is the expected number of points you would earn?

number of heads, probability and points

\

$$\begin{tabular}{|c|c|c|c|c|} \\hline number of heads & 0 & 1 & 2 & 3 \\\\ \\hline probability & \\frac{1}{8} & \\frac{3}{8} & \\frac{3}{8} & \\frac{1}{8} \\\\ \\hline points & 0 & 2 & 5 & 10 \\\\ \\hline \\end{tabular}$$

• 0.375
• 4.625
• 3.875
• 0.333

Explanation:

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<reasoning>

Define the expected value formula

\[
E(X) = \sum x_i \cdot P(X = x_i)
\]

Substitute the given points and probabilities

\[
E(X) = 0 \cdot \frac{1}{8} + 2 \cdot \frac{3}{8} + 5 \cdot \frac{3}{8} + 10 \cdot \frac{1}{8}
\]

Calculate the final expected value

\[

$$\begin{aligned} E(X) &= 0 + \frac{6}{8} + \frac{15}{8} + \frac{10}{8} \\ &= \frac{31}{8} \\ &= 3.875 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) 0.375</mcq-option>
<mcq-option>(B) 4.625</mcq-option>
<mcq-correct>(C) 3.875</mcq-correct>
<mcq-option>(D) 0.333</mcq-option>
</answer>

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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
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Answer:

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"step_cot",
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</pre_analysis>

<reasoning>

Define the expected value formula

\[
E(X) = \sum x_i \cdot P(X = x_i)
\]

Substitute the given points and probabilities

\[
E(X) = 0 \cdot \frac{1}{8} + 2 \cdot \frac{3}{8} + 5 \cdot \frac{3}{8} + 10 \cdot \frac{1}{8}
\]

Calculate the final expected value

\[

$$\begin{aligned} E(X) &= 0 + \frac{6}{8} + \frac{15}{8} + \frac{10}{8} \\ &= \frac{31}{8} \\ &= 3.875 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) 0.375</mcq-option>
<mcq-option>(B) 4.625</mcq-option>
<mcq-correct>(C) 3.875</mcq-correct>
<mcq-option>(D) 0.333</mcq-option>
</answer>

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