Chapter 10.4, Problem 19E

Solution

a)

To calculate: The probability that a basketball player makes exactly 5 of 6 foul shots.

The probability that a basketball player makes exactly 5 of 6 foul shots is 0.3932.

Given:

The percentage of foul shots a basketball player can make $\left(p\right)=80\%=0.80$ .

The total number of foul shots during a game $\left(n\right)=6$

Concept used:A probability distribution where each trial has two possible outcomes known as a success and failure, each trial is independent and the probability of each trial is the same, then to calculate the probability of a certain number of successes occurring, a binomial probability distribution is used as shown below

  $P(X=k)=\left(\begin{array}{c}n\\ k\end{array}\right)p^k{(1-p)}^{n-k}$

where k is the number of successes, n is the number of trials and p is the probability of success.

Calculation:

The probability is calculated as:

  $\begin{array}{c}$ P\left(X=5\right)=\left(\begin{array}{c}6\\ 5\end{array}\right){(0.8)}^5{(1-0.8)}^{6-5}$=\frac{6!}{5!\left(6-5\right)!}\times {0.8}^5\times {0.2}^1\\ =0.3932\end{array}$

Conclusion:The probability that a basketball player makes 5 out of 6 foul shots in a game is about 0.3932.

b)

To calculate: The probability that a basketball player makes no more than 2 out of 6 foul shots.

The probability that a basketball player makes no more than 2 of 6 foul shots is 0.0170.

Calculation:

The probability is calculated as:

  $\begin{array}{c}P\left(X\le 2\right)=P(X=0)+P(X=1)+P(X=2)\\ $ =\left(\begin{array}{c}6\\ 0\end{array}\right){(0.8)}^0{(1-0.8)}^{6-0}+\left(\begin{array}{c}6\\ 1\end{array}\right){(0.8)}^1{(1-0.8)}^{6-1}+\left(\begin{array}{c}6\\ 2\end{array}\right){(0.8)}^2{(1-0.8)}^{6-2}$=\frac{6!}{0!\left(6-0\right)!}\times {0.8}^0\times {0.2}^6+\frac{6!}{1!\left(6-1\right)!}\times {0.8}^1\times {0.2}^5+\frac{6!}{2!\left(6-2\right)!}\times {0.8}^2\times {0.2}^4\\ =0.01696\end{array}$

Conclusion:The probability that a basketball player makes no more than 2 out of 6 foul shots in a game is about 0.0170 (rounded to 4 decimal places).