Question

The probability that a certain kind of component will survive a given shock test is $\frac{3}{4}$ The probability that exactly $2$ of the next $4$ components tested survive is

• A. $\frac{9}{41}$
• B. $\frac{25}{128}$
• C. $\frac{1}{5}$
• D. $\frac{27}{128}$

Answer 1

The correct option is D

$\frac{27}{128}$

The explanation for the correct answer:

Solve for the probability that exactly $2$of the next $4$ components tested survived:

Assume that ‘$p$’ is the probability of a certain kind of component.

We are given, $p=\frac{3}{4}$

$q=1-p,p=1-\frac{3}{4}=\frac{1}{4}$

We have, $n=4$

The binomial distribution is,

$b(x;n,p)={}^{n}C_x\times p^x\times {(1–p)}^{n–x}$

$P($Exactly $2$ of next $4$ components$)=P\left(X=2\right)$

$$\begin{gathered} P(X=2)={}^{4}C_2{\left(\frac{3}{4}\right)}^2{\left(\frac{1}{4}\right)}^2 \\ P(X=2)=6\times \left(\frac{9}{16}\right)\left(\frac{1}{16}\right) \\ P(X=2)=\frac{54}{256} \\ P(X=2)=\frac{27}{128} \end{gathered}$$

Hence, option(D) is the correct answer.