Chapter 12.7, Problem 22PPS

a.

To determine

Graph the binomial probability distribution.

Answer to Problem 22PPS

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Explanation of Solution

Given information:

A beverage company has a promotion in which $30\%$ of the bottles purchased during the promotion have bottle caps that win a free beverage. Melanie bought $10$ bottles.

Calculation:

Table of probabilities for all possible value of $x$ .

  $\begin{array}{cccccccccccc}X& 0& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10\\ P\left(X=x\right)& \begin{array}{l}0.0282\\ 4752\end{array}& \begin{array}{l}0.121\\ 06082\end{array}& \begin{array}{l}0.2334\\ 7444\end{array}& \begin{array}{l}0.266\\ 82793\end{array}& \begin{array}{l}0.2001\\ 2095\end{array}& \begin{array}{l}0.1029\\ 1935\end{array}& \begin{array}{l}0.0367\\ 5691\end{array}& \begin{array}{l}0.009\\ 00169\end{array}& \begin{array}{l}0.001\\ 4467\end{array}& \begin{array}{l}0.000\\ 13778\end{array}& \begin{array}{l}0.000\\ 0059\end{array}\end{array}$

  

Hence the graph represents the binomial probability distribution.

b.

To determine

Find the probability that Melanie won at least $4$ free beverages.

Answer to Problem 22PPS

  $0.3504$

Explanation of Solution

Given information:

A beverage company has a promotion in which $30\%$ of the bottles purchased during the promotion have bottle caps that win a free beverage. Melanie bought $10$ bottles.

Calculation:

The probability that Melanie won at least $4$ free beverages,

  $\begin{array}{l}P\left(X\ge 4\right)=1-P\left(X\le 3\right)\\ =1-{\displaystyle \sum _{x=0}^3\left(\underset{x}{\overset{3}{{\displaystyle }}}\right)}{\left(0.30\right)}^x{\left(0.70\right)}^{10-x}\\ =0.3504\end{array}$

Hence the probability is $0.3504$ .