Question

the pointer shown to the right can land on every number, and the respective probability that the pointer can land on is shown in the table below. compute the expected value for the number on which the pointer lands. the expected value for the number on which the pointer lands is \\(\square\\). (type an integer or a decimal.) outcome probability 1 \\(\frac{1}{8}\\) 2 \\(\frac{1}{4}\\) 3 \\(\frac{1}{8}\\) 4 \\(\frac{1}{2}\\)

Explanation:

Step1: Recall the formula for expected value

The expected value \( E(X) \) of a discrete random variable is calculated as \( E(X)=\sum_{i} x_i P(x_i) \), where \( x_i \) are the outcomes and \( P(x_i) \) are their respective probabilities.

Step2: Identify the outcomes and their probabilities

From the table:

• Outcome \( x_1 = 1 \), Probability \( P(x_1)=\frac{1}{8} \)
• Outcome \( x_2 = 2 \), Probability \( P(x_2)=\frac{1}{4} \)
• Outcome \( x_3 = 3 \), Probability \( P(x_3)=\frac{1}{8} \)
• Outcome \( x_4 = 4 \), Probability \( P(x_4)=\frac{1}{2} \)

Step3: Calculate each term \( x_i P(x_i) \)

• For \( x_1 = 1 \): \( 1\times\frac{1}{8}=\frac{1}{8} = 0.125 \)
• For \( x_2 = 2 \): \( 2\times\frac{1}{4}=\frac{2}{4}=0.5 \)
• For \( x_3 = 3 \): \( 3\times\frac{1}{8}=\frac{3}{8}=0.375 \)
• For \( x_4 = 4 \): \( 4\times\frac{1}{2}=\frac{4}{2} = 2 \)

Step4: Sum up all the terms

\( E(X)=0.125 + 0.5+0.375 + 2 \)
First, add \( 0.125+0.5 = 0.625 \)
Then, \( 0.625+0.375 = 1 \)
Finally, \( 1 + 2=3 \)

Answer:

3