Chapter 12.6, Problem 53SR

To determine

To find: Probability that there will be $3$ failures.

Answer to Problem 53SR

Probability that there will be $3$ failures is about $0.267$ .

Explanation of Solution

Given information:

Rate of success $=70\%$

It means

Probability of success $=0.7$

Total number of trails $=n=10$

Number of failures $=3$

Concept:

Formula of Binomial distribution is

  $b\left(x;n,P\right)=C{}_x\times P^x\times {\left(1-P\right)}^{n-x}$

Where

  $b=$ Binomial probability

  $x=$ total number of “successes”.

  $P=$ probability of a success on an individual trial.

  $n=$ Number of trials.

Given

  $\begin{array}{l}n=10\\ P=0.7\end{array}$

Number of Failures $=3$

It means

Number of success $=10-3=7$

Therefore,

  $x=7$

Therefore,

On substituting values in above formula

We get

  $\begin{array}{l}b\left(x;n,P\right)=C{}_x\times P^x\times {\left(1-P\right)}^{n-x}\\ \Rightarrow b=C{}_7\times {\left(0.7\right)}^7\times {\left(1-0.7\right)}^{10-7}\\ \Rightarrow b=\frac{10!}{\left(10-7\right)!7!}\times {\left(0.7\right)}^7\times {\left(0.3\right)}^3\\ \Rightarrow b=\frac{10\times 9\times 8\times 7!}{\left(3\right)!7!}\times \left(0.08235\right)\times \left(0.027\right)\\ \Rightarrow b=\frac{10\times 9\times 8}{6}\times \left(0.08235\right)\times \left(0.027\right)\\ \Rightarrow b=\frac{720}{6}\times \left(0.00222345\right)\\ \Rightarrow b=0.267\end{array}$

Therefore,

Probability that there will be $3$ failures is about $0.267$ .