Let's break this down step-by-step.
### Part (a): Calculating the Expected Value for Each Choice
Choice 1: Passing the Ball
1. We are given the values ([tex]\(x_i\)[/tex]) and their corresponding probabilities ([tex]\(P(x_i)\)[/tex]) for passing the ball:
- [tex]\(x_i = 3\)[/tex], [tex]\(P(x_i) = 0.30\)[/tex]
- [tex]\(x_i = 0\)[/tex], [tex]\(P(x_i) = 0.70\)[/tex]
2. The formula for the expected value [tex]\(E\)[/tex] is given by:
[tex]\[
E = \sum (x_i \cdot P(x_i))
\][/tex]
3. Substituting the given values into the formula, we have:
[tex]\[
E_{\text{passing}} = (3 \cdot 0.30) + (0 \cdot 0.70)
\][/tex]
4. Simplifying this:
[tex]\[
E_{\text{passing}} = 0.90
\][/tex]
Choice 2: Taking the Shot
1. We are given the values ([tex]\(x_i\)[/tex]) and their corresponding probabilities ([tex]\(P(x_i)\)[/tex]) for taking the shot:
- [tex]\(x_i = 2\)[/tex], [tex]\(P(x_i) = 0.48\)[/tex]
- [tex]\(x_i = 0\)[/tex], [tex]\(P(x_i) = 0.52\)[/tex]
2. Using the same formula for the expected value [tex]\(E\)[/tex]:
[tex]\[
E_{\text{shooting}} = \sum (x_i \cdot P(x_i))
\][/tex]
3. Substituting the given values into the formula, we have:
[tex]\[
E_{\text{shooting}} = (2 \cdot 0.48) + (0 \cdot 0.52)
\][/tex]
4. Simplifying this:
[tex]\[
E_{\text{shooting}} = 0.96
\][/tex]
### Part (b): Decision Making
To determine whether to pass the ball or take the shot, we compare the expected values calculated above:
- Expected value for passing the ball: [tex]\(E_{\text{passing}} = 0.90\)[/tex]
- Expected value for taking the shot: [tex]\(E_{\text{shooting}} = 0.96\)[/tex]
Since the expected value for taking the shot ([tex]\(0.96\)[/tex]) is greater than the expected value for passing the ball ([tex]\(0.90\)[/tex]), the player should take the shot themselves.
### Conclusion
- Expected value for passing the ball: [tex]\(0.90\)[/tex]
- Expected value for taking the shot: [tex]\(0.96\)[/tex]
- Decision: The player should take the shot because the expected value of taking the shot is higher than that of passing the ball.
