Question

of all the people applying for a certain job, 60% are qualified and 40% are not. the personnel manager claims that she approves qualified people 80% of the time; she approves unqualified people 40% of the time. find the probability that a person is not qualified if the person was approved by the manager.
the probability is
(type an integer or decimal rounded to four decimal places as needed )

Explanation:

Step1: Define the events

Let $Q$ be the event that a person is qualified, $\overline{Q}$ be the event that a person is not - qualified, and $A$ be the event that a person is approved.
We know that $P(Q)=0.6$, $P(\overline{Q}) = 0.4$, $P(A|Q)=0.8$, and $P(A|\overline{Q})=0.4$.

Step2: Calculate $P(A)$ using the law of total - probability

By the law of total - probability, $P(A)=P(A|Q)P(Q)+P(A|\overline{Q})P(\overline{Q})$.
Substitute the known values: $P(A)=(0.8\times0.6)+(0.4\times0.4)=0.48 + 0.16=0.64$.

Step3: Calculate $P(\overline{Q}|A)$ using Bayes' theorem

By Bayes' theorem, $P(\overline{Q}|A)=\frac{P(A|\overline{Q})P(\overline{Q})}{P(A)}$.
Substitute the values we found: $P(\overline{Q}|A)=\frac{0.4\times0.4}{0.64}=\frac{0.16}{0.64}=0.2500$.

Answer:

$0.2500$