players in a card game use the spinner below to determine how many cards they will pick up during their…

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})).
2. 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).

Explanation:

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