Question

At a production process, the produced items are tested for defects. A defective unit is classified as such with probability 0.9, whereas a correct unit is classified as such with probability 0.85. Furthermore, 10% of the produced units are defective. What is the conditional probability that a unit is correct, given that is has been classified as defective?

Answer 1

Let $A$ be the event that an item is defective(correct) and $B$ be the event that an item is classified defective.

The conditional probability that we need to find is $p(A|B)$.

We are given that,

$p(A)=0.1\implies p(A')=1-p(A)=1-0.1=0.9$

$p(B|A)=0.9$ and $p(B|A')=1-P(B'|A')=1-0.85=0.15$

To find $p(B)$, we use the law of total probability as follows.

$P(B) = p(A)\times p(B|A) + p(A')\times p(B|A')=(0.1\times 0.9)+(0.9\times0.15)=0.225$

Since $p(A)\times p(B|A) = p(B)\times p(A|B)$, we substitute for the values we have obtained above to get $p(A|B)$.

Now,

$p(A|B)={p(A)\times p(B|A)\over p(B)}={0.1\times0.9\over 0.225}=0.4$

Therefore, the conditional probability that a unit is defective(correct), given that it has been classified as defective is 0.4.