Question

1. the point guard of a basketball team has to make a decision about whether or not to shoot a three - point attempt or pass the ball to another player who will shoot a two - point shot. the point guard makes three - point shots 30 percent of the time, while his teammate makes the two - point shot 48 percent of the time.

a. what is the expected value for each choice?
b. should he pass the ball or take the shot himself? explain.

Explanation:

Step1: Calculate the expected value for shooting a three - point shot

The formula for the expected value \(E(X)=\sum_{i}x_{i}P(x_{i})\). For the three - point shot, \(x_1 = 3\) with \(P(x_1)=0.30\) and \(x_2 = 0\) with \(P(x_2)=0.70\).

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Step2: Calculate the expected value for passing (two - point shot)

For the two - point shot, \(x_1 = 2\) with \(P(x_1)=0.48\) and \(x_2 = 0\) with \(P(x_2)=0.52\)

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Step3: Compare the expected values

We have \(E(X_{3 - point})=0.9\) and \(E(X_{2 - point}) = 0.96\)

Answer:

a. The expected value for shooting a three - point shot is \(0.9\), and the expected value for passing (two - point shot) is \(0.96\).
b. He should pass the ball. Since the expected value of the two - point shot (\(0.96\)) is greater than the expected value of the three - point shot (\(0.9\)), passing the ball gives a higher average number of points in the long run.