Chapter 11.4, Problem 25PPS

To determine

What is the probability that there will be at $12$ failures?

Answer to Problem 25PPS

  $\boxed{0.0145or1.45\%}$

Explanation of Solution

Given information:

A binomial distribution has a $60\%$ rate of success. There are $18$ trials. What is the probability that there will be at $12$ failures?

Calculation:

Generally, the probability of $k$ successes in $n$ trials of a binomial experiment is,

  $P(k){=}^n{C}_k{(p)}^k{(q)}^{n-k}$

Here $p$ represents the probability of success and $q$ represents the probability of failure.

Examine that a binomial distribution has $60\%$ rate of success and the total number of trials is $18$ .

So, the probability of success is,

  $\begin{array}{l}p=60\%\\ =0.6\end{array}$

The probability of failure is,

  $\begin{array}{l}q=1-p\\ =1-0.6\\ =0.4\end{array}$

So, the probability that there will be $12$ failures.

So, the number of successes is $18-12=6$ .

Substitute $p=0.6,q=0.4,k=6,n=18$ in the above formula,

  $\begin{array}{l}P\left(6\right){=}^{18}{C}_6{(}^{0.6}{(}^{0.4}\\ =18,564{(}^{0.6}{(}^{0.4}\\ =0.0145\\ =1.45\%\end{array}$

Hence, the probability that there will be $12$ failures is $\boxed{0.0145or1.45\%}$ .