Question

there are 4 different numbers in a hat, with many duplicates of each number. the probability distribution for each number is below. calculate the expected value.

number and probability

\

$$\begin{tabular}{|c|c|} \\hline number & probability \\\\ \\hline 3 & 0.1 \\\\ \\hline 6 & 0.25 \\\\ \\hline 9 & 0.3 \\\\ \\hline 12 & 0.35 \\\\ \\hline \\end{tabular}$$

• 7.5
• 1
• 8.7
• 0.25

Explanation:

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<reasoning>

Define the expected value formula

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

Substitute the given values into the formula

\[
E(X) = (3 \cdot 0.1) + (6 \cdot 0.25) + (9 \cdot 0.3) + (12 \cdot 0.35)
\]

Calculate the final expected value

\[

$$\begin{aligned} E(X) &= 0.3 + 1.5 + 2.7 + 4.2 \\ E(X) &= 8.7 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) 7.5</mcq-option>
<mcq-option>(B) 1</mcq-option>
<mcq-correct>(C) 8.7</mcq-correct>
<mcq-option>(D) 0.25</mcq-option>
</answer>

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Answer:

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<reasoning>

Define the expected value formula

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

Substitute the given values into the formula

\[
E(X) = (3 \cdot 0.1) + (6 \cdot 0.25) + (9 \cdot 0.3) + (12 \cdot 0.35)
\]

Calculate the final expected value

\[

$$\begin{aligned} E(X) &= 0.3 + 1.5 + 2.7 + 4.2 \\ E(X) &= 8.7 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) 7.5</mcq-option>
<mcq-option>(B) 1</mcq-option>
<mcq-correct>(C) 8.7</mcq-correct>
<mcq-option>(D) 0.25</mcq-option>
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