Chapter 8, Problem 8.71RE

(a)

To determine

To show that the setting in the question is binomial setting.

Explanation of Solution

Given:

  $\begin{array}{l}n=7\\ p=0.80\end{array}$

We know that the Binomial theorem is as:

  $P(X=k)=C{}_k\times p^k\times {(1-p)}^{n-k}$

The four conditions for a binomial setting are: binary (success/failure), independent trials, fixed number of trials and the probability of success is the same for each trial.

Binary: Satisfied because success is “make the shot” and failure is “fail to make a shot”.

Independent trials: Satisfied, assuming that the different free throws are independent.

Fixed number of trials:Satisfied, because the fixed number of trials are seven free throws.

Probability of Success: Satisfied, because the probability of success is $p=80\%$

Thus the setting is binomial.

(b)

To determine

To calculate the probability that she makes four or fewer baskets on seven free throws.

Answer to Problem 8.71RE

The probability is $0.1480$ .

Explanation of Solution

Given:

  $\begin{array}{l}n=7\\ p=0.80\end{array}$

We know that the Binomial theorem is as:

  $P(X=k)=C{}_k\times p^k\times {(1-p)}^{n-k}$

Evaluate at $k=0,1,2,3,4$ :

  $\begin{array}{}$ P(X=0)=C{}_0\times {(0.8)}^0\times {(1-0.8)}^{7-0}\approx 0.0000128$$ P(X=1)=C{}_1\times {(0.8)}^1\times {(1-0.8)}^{7-1}\approx 0.0003584$$ P(X=2)=C{}_2\times {(0.8)}^2\times {(1-0.8)}^{7-2}\approx 0.0043008$$ P(X=3)=C{}_3\times {(0.8)}^3\times {(1-0.8)}^{7-3}\approx 0.028672$$ P(X=4)=C{}_4\times {(0.8)}^4\times {(1-0.8)}^{7-4}\approx 0.114688$\end{array}$

Use the addition rule:

  $\begin{array}{l}P(X\le 4)=P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)\\ =0.0000128+0.0003584+0.0043008+0.028672+0.114688\\ \approx 0.1480\end{array}$

Hence the result.

(c)

To determine

To explain would you be suspicious about her claim of being an $80\%$ free throw shooter.

Answer to Problem 8.71RE

No.

Explanation of Solution

No, you should not be suspicious about her claim of being an $80\%$ free throw shooter because the probability of making four free throws or less is not small thus she can make it. Thus you should not be suspicious of her being claiming.