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:

1. First, identify the values and their probabilities from the spinner and table:

• From the spinner, we can see that the number of cards \(x\) can be 1, 2, or 3. And from the table (assuming the full - probability distribution is: when \(x = 1\), \(P(x = 1)=0.5\)). We need to find the probabilities for \(x = 2\) and \(x = 3\). There are 8 sections on the spinner. The number of sections with 1 is 4, with 2 is 3, and with 3 is 1. So \(P(x = 2)=\frac{3}{8}=0.375\) and \(P(x = 3)=\frac{1}{8}=0.125\).
• The formula for the expected value \(E(X)\) of a discrete - random variable is \(E(X)=\sum_{i}x_{i}P(x_{i})\).

1. Then, calculate the expected value:

• For \(x = 1\) and \(P(x = 1)=0.5\), the product is \(1\times0.5 = 0.5\).
• For \(x = 2\) and \(P(x = 2)=0.375\), the product is \(2\times0.375 = 0.75\).
• For \(x = 3\) and \(P(x = 3)=0.125\), the product is \(3\times0.125 = 0.375\).
• Now, sum up these products: \(E(X)=1\times0.5 + 2\times0.375+3\times0.125\).
• \(E(X)=0.5 + 0.75+0.375\).
• \(E(X)=1.625\).

Step1: Determine probabilities

The spinner has 8 sections. 4 for 1, 3 for 2, 1 for 3. So \(P(1) = 0.5\), \(P(2)=0.375\), \(P(3)=0.125\).

Step2: Apply expected - value formula

\(E(X)=\sum_{i}x_{i}P(x_{i})=1\times0.5 + 2\times0.375+3\times0.125\)

Step3: Calculate the sum

\(E(X)=0.5 + 0.75+0.375 = 1.625\)

Answer:

1.625