Question

players in a card game use the spinner below to determine how many cards they will pick up during their turn. the table below shows the probability distribution for the number of cards a player will pick up during one turn. what is the expected value for the number of cards a player will pick up during one turn?

Explanation:

Step1: Recall the formula for expected value

The expected value \( E(X) \) of a discrete random variable is calculated by multiplying each possible value \( x_i \) by its corresponding probability \( P(x_i) \) and then summing these products. The formula is \( E(X)=\sum_{i} x_i \cdot P(x_i) \).

Step2: Identify the values and their probabilities

From the table, we have:

• When \( x_1 = 1 \), \( P(x_1)=0.5 \)
• When \( x_2 = 2 \), \( P(x_2)=0.3 \)
• When \( x_3 = 3 \), \( P(x_3)=0.2 \)

Step3: Calculate each product \( x_i \cdot P(x_i) \)

• For \( x_1 = 1 \) and \( P(x_1)=0.5 \): \( 1\times0.5 = 0.5 \)
• For \( x_2 = 2 \) and \( P(x_2)=0.3 \): \( 2\times0.3 = 0.6 \)
• For \( x_3 = 3 \) and \( P(x_3)=0.2 \): \( 3\times0.2 = 0.6 \)

Step4: Sum the products

Now, we sum these results: \( E(X)=0.5 + 0.6+0.6 \)
\( E(X)=1.7 \)

Answer:

1.7