Certainly! Let's solve the question step-by-step.
### Part (a): Calculate the Expected Values
The expected value is calculated as the sum of all possible outcomes, each multiplied by their respective probabilities. We need to determine the expected value for two scenarios: the point guard taking a three-point shot and the teammate taking a two-point shot.
#### Expected Value for the Point Guard's Three-Point Shot
The outcomes and probabilities for the point guard’s three-point shot are as follows:
- Makes a three-point shot (outcome = 3 points) with a probability of 0.30.
- Misses the three-point shot (outcome = 0 points) with a probability of 0.70.
To calculate the expected value:
[tex]\[ \text{Expected Value}_{\text{three-point}} = (3 \times 0.30) + (0 \times 0.70) \][/tex]
[tex]\[ \text{Expected Value}_{\text{three-point}} = 0.90 \][/tex]
#### Expected Value for the Teammate's Two-Point Shot
The outcomes and probabilities for the teammate’s two-point shot are as follows:
- Makes a two-point shot (outcome = 2 points) with a probability of 0.48.
- Misses the two-point shot (outcome = 0 points) with a probability of 0.52.
To calculate the expected value:
[tex]\[ \text{Expected Value}_{\text{two-point}} = (2 \times 0.48) + (0 \times 0.52) \][/tex]
[tex]\[ \text{Expected Value}_{\text{two-point}} = 0.96 \][/tex]
So the expected values are:
- Expected value for the three-point shot by the point guard: 0.90 points
- Expected value for the two-point shot by the teammate: 0.96 points
### Part (b): Decision Making
To determine whether the point guard should take the three-point shot or pass the ball to the teammate for the two-point shot, we compare the expected values calculated above:
- The expected value for the point guard’s three-point shot is 0.90.
- The expected value for the teammate’s two-point shot is 0.96.
Since the expected value for the teammate's shot (0.96) is higher than the expected value for the point guard's shot (0.90), the point guard should:
Pass the ball to the teammate.
This decision is based on the higher expected value, which suggests that passing the ball to the teammate provides a better chance of scoring more points.
