\begin{tabular}{|c|c|c|}
\hline
[tex]$x_i$[/tex] & 3 & 0 \\
\hline
[tex]$P\left(x_i\right)$[/tex] & 0.30 & 0.70 \\
\hline
\end{tabular}
\begin{tabular}{|c|c|c|}
\hline
[tex]$x_i$[/tex] & 2 & 0 \\
\hline
[tex]$P\left(x_i\right)$[/tex] & 0.48 & 0.52 \\
\hline
\end{tabular}

a. What is the expected value for each choice?

b. Should he pass the ball or take the shot himself? Explain.

Claire is considering investing in a new business. In the first year, there is a probability of 0.2 that...



Answer :

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.