Chapter 10.6, Problem 13E

(a)

To determine

To graph:

The histogram of the binomial distribution for the survey

Answer to Problem 13E

The Histograph of the binomial distribution for the survey are as shown:

  

Histograph

Explanation of Solution

Given information:

  

The probability distribution is given as:

  $\begin{array}{l}n=6\\ p=0.27\\ k=0,1,2,\cdot \cdot \cdot ,6\end{array}$

Interpretation:

The binomial experiment, the probability of k successes in number of trial is:

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

Put the values $n=6,p=0.27,k=0,1,2,\cdot \cdot \cdot ,6$ in equation (1):

  $\begin{array}{l}P\left(k=0\right)=C{}_0{\left(0.27\right)}^0{\left(0.73\right)}^7\\ P\left(k=0\right)=0.1513\\ P\left(k=1\right)=C{}_1{\left(0.27\right)}^1{\left(0.73\right)}^6\\ P\left(k=1\right)=0.3358\\ P\left(k=2\right)=C{}_2{\left(0.27\right)}^2{\left(0.73\right)}^5\\ P\left(k=2\right)=0.3105\\ P\left(k=3\right)=C{}_3{\left(0.27\right)}^3{\left(0.73\right)}^4\\ P\left(k=3\right)=0.1531\\ P\left(k=4\right)=C{}_4{\left(0.27\right)}^4{\left(0.73\right)}^3\\ P\left(k=4\right)=0.0425\\ P\left(k=5\right)=C{}_5{\left(0.27\right)}^5{\left(0.73\right)}^2\\ P\left(k=5\right)=0.0063\\ P\left(k=6\right)=C{}_6{\left(0.27\right)}^6{\left(0.73\right)}^1\\ P\left(k=6\right)=0.0005\end{array}$

The probability value of $x$ is

  $\begin{array}{l}P\left(k=0\right)=0.1513,P\left(k=1\right)=0.3358,P\left(k=2\right)=0.3105,P\left(k=3\right)=0.1531,P\left(k=4\right)=0.0425,P\left(k=5\right)=0.0063,\\ P\left(k=6\right)=0.0005\end{array}$ The Histograph of the binomial distribution for the survey are as shown:

  

Histograph

(b)

To determine

To calculate:

The most possible outcomes of the survey

Answer to Problem 13E

The most possible outcome of the survey is $2$ .

Explanation of Solution

Given information:

  

Calculation:

The possible outcome of this survey is the value of $x$ for that $P\left(x\right)$ is higher.

The probability distribution result shows the probability is great for $2$ , nearly $0.34$ .

The most possible outcome of the survey is $2$ .

(c)

To determine

To calculate:

The probability at most $2$ people have the class ring.

Answer to Problem 13E

The probability of at most $2$ people have the class ring is $0.7976$ .

Explanation of Solution

Given information:

  

Calculation:

The required probability distribution is:

  $P\left(k\le 2\right)=P\left(k=0\right)+P\left(k=1\right)+P\left(k=2\right)$

Put the values $P\left(k=0\right)=0.1513,P\left(k=1\right)=0.3358,P\left(k=2\right)=0.3105$ in equation

  $\begin{array}{l}P\left(k\le 2\right)=0.1513+0.3358+0.3105\\ P\left(k\le 2\right)=0.7976\end{array}$

The probability of at most $2$ people have the class ring is $0.7976$ .