Question

a spinner has 4 numbers on it, all with an equal amount of space. the table shows the probabilities for each number. calculate the expected value.

number and probability
\

$$\begin{tabular}{|c|c|} \\hline number & probability \\\\ \\hline 1 & \\frac{1}{4} \\\\ \\hline 2 & \\frac{1}{4} \\\\ \\hline 3 & \\frac{1}{4} \\\\ \\hline 4 & \\frac{1}{4} \\\\ \\hline \\end{tabular}$$

• 1
• 3
• 2.5
• 2

Explanation:

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<reasoning>

Identify the values and probabilities

\[

$$\begin{aligned} &x_1 = 1, \quad P(x_1) = \frac{1}{4} \\ &x_2 = 2, \quad P(x_2) = \frac{1}{4} \\ &x_3 = 3, \quad P(x_3) = \frac{1}{4} \\ &x_4 = 4, \quad P(x_4) = \frac{1}{4} \end{aligned}$$

\]

Apply the expected value formula

\[

$$\begin{aligned} E(X) &= \sum_{i=1}^{4} x_i \cdot P(x_i) \\ E(X) &= 1 \cdot \frac{1}{4} + 2 \cdot \frac{1}{4} + 3 \cdot \frac{1}{4} + 4 \cdot \frac{1}{4} \end{aligned}$$

\]

Calculate the final sum

\[

$$\begin{aligned} E(X) &= \frac{1 + 2 + 3 + 4}{4} \\ E(X) &= \frac{10}{4} = 2.5 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>1</mcq-option>
<mcq-option>3</mcq-option>
<mcq-correct>2.5</mcq-correct>
<mcq-option>2</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the values and probabilities

\[

$$\begin{aligned} &x_1 = 1, \quad P(x_1) = \frac{1}{4} \\ &x_2 = 2, \quad P(x_2) = \frac{1}{4} \\ &x_3 = 3, \quad P(x_3) = \frac{1}{4} \\ &x_4 = 4, \quad P(x_4) = \frac{1}{4} \end{aligned}$$

\]

Apply the expected value formula

\[

$$\begin{aligned} E(X) &= \sum_{i=1}^{4} x_i \cdot P(x_i) \\ E(X) &= 1 \cdot \frac{1}{4} + 2 \cdot \frac{1}{4} + 3 \cdot \frac{1}{4} + 4 \cdot \frac{1}{4} \end{aligned}$$

\]

Calculate the final sum

\[

$$\begin{aligned} E(X) &= \frac{1 + 2 + 3 + 4}{4} \\ E(X) &= \frac{10}{4} = 2.5 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>1</mcq-option>
<mcq-option>3</mcq-option>
<mcq-correct>2.5</mcq-correct>
<mcq-option>2</mcq-option>
</answer>

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