Chapter 10, Problem 17CR

To determine

To calculate:

Probability distribution of the survey and draw a histogram.

Answer to Problem 17CR

The probability distribution of the survey result shows that the probability is greatest for $4$ which is nearly $0.4$ .

Explanation of Solution

Given information:

Basket ball player makes a free throw $82.6\%$ of the time.

Player attempt $5$ free attempts.

For binomial experiment, the probability of exactly $k$ successes in $n$ trials as given:

  $P\left(ksuccess\right)=C{}_k{p}^k{\left(1-p\right)}^{n-k}$

Where,

  $n=5p=0.826,k=0,1,\mathrm{2...5}$

Now substitute the values in stated formula and get required probability.

  $\begin{array}{l}P\left(k=0\right)=C{}_0{\left(0.826\right)}^0{\left(0.174\right)}^5\\ =0.0002\\ P\left(k=1\right)=C{}_1{\left(0.826\right)}^1{\left(0.174\right)}^4\\ =0.0038\\ P\left(k=2\right)=C{}_1{\left(0.826\right)}^2{\left(0.174\right)}^3\\ =0.0359\\ P\left(k=3\right)=C{}_1{\left(0.826\right)}^3{\left(0.174\right)}^2\\ =0.1706\\ P\left(k=4\right)=C{}_1{\left(0.826\right)}^4{\left(0.174\right)}^1\\ =0.4050\\ P\left(k=5\right)=C{}_1{\left(0.826\right)}^5{\left(0.174\right)}^0\\ =0.3845\end{array}$

Histogram which shows a probability distribution of the random variable is as follows:

  

Above histogram shows the probability distribution of free throws. The most likely outcome of the survey is the value of $x$ for which $P\left(x\right)$ is highest.

Hence, the probability distribution of the survey result shows that the probability is greatest for $4$ which is nearly $0.4$ .