Q.4.81 - A First Course in Probability

Question

In Example $8i$, what percentage of $i$ defective lots does the purchaser reject? Find it for$i = 1, 4.$ Given that a lot is rejected, what is the conditional probability that it contained $4$defective components?

Step-by-Step Solution

Answer

$P(4$ defective$\mid$lot rejected $)=\frac{75}{138}$

Step 1 : Given information

Given in the question that we need to find the conditional probability that it contained $4$defective components 

Step 2 : Explanation

Let$i$ be the defective lot.

The number of components in a lot is $10.$

It is known that the $3$components are randomly selected for inspection, if all $3$ components are non-defective then lot will be accepted.

Find the probability that lot $1$ is non-defective or accepted.

Here,$i=1$ lot has $1$ defective.

So,

$P($Accepted $)=\frac{\left(\begin{array}{l}1\\ 0\end{array}\right)\left(\begin{array}{l}9\\ 3\end{array}\right)}{\left(\begin{array}{l}10\\ 3\end{array}\right)}$

$=\frac{7}{10}$

Find the probability that lot $1$ is defective.

Therefore,

$P($ rejected $)=1-P($ Accepted$)$

 $=1-\frac{7}{10}$

$=\frac{3}{10}$

Step 3 : Explanation

When $i=4$ implies that lot has 4 defective components.

Now,

$\mathrm{P}($ accepted $)=$$\frac{\left(\begin{array}{l}4\\ 0\end{array}\right)\left(\begin{array}{l}6\\ 3\end{array}\right)}{\left(\begin{array}{l}10\\ 3\end{array}\right)}$$=\frac{1}{6}$

Find the probability that lot $4$ is defective.

Therefore,

$P($ rejected $)=1-\frac{1}{6}$

$=\frac{5}{6}$

Step 4 : Conditional probability

Find the conditional probability that it contained$4$ defective components given that lot 1 and 4 are rejected.

Therefore, the required probability is,

$P(4$ defective $\mid$lot rejected$)$ $=\frac{\left(\frac{5}{6}\times \frac{3}{10}\right)}{\left(\frac{5}{6}\times \frac{3}{10}\right)+\left(\frac{3}{10}\times \frac{7}{10}\right)}$

$=\frac{75}{138}$

Step 5 : Final answer

Required probability  

$P(4$ defective $\mid$ lot rejected$)$$=\frac{75}{138}$