Conditional Probability Formula Confusion

Question

Of eight otherwise equal candidates for a job for a position in the Planning Department, three are qualified accountants, four are financial planners and two have neither of these qualifications. Human Resources is interested in the outcome as part of their workforce planning review. They want to know:

• The probability that a financial planner got the job
• Given that a qualified accountant got the job, the probability that s/he is a also a financial planner
• The probability that a qualified accountant got the job, given that a financial planner did not get the job.

So for this question, the solutions say to draw up a table and fill in the information or use the formula P(A|B)= P(A and B)/P(B). I can find the solution if I fill in the table but the formula isn't working out for me... I don't understand it. If P(A|B)= P(A and B)/P(B), then wouldn't P(A|B) = P(B)?

The answers given are 4/8 for the first one (don't have a problem with this), 1/3 for the second, and 2/4 for the last one.

The formula P(A|B)= P(A and B)/P(B), where P(A|B) means P(A), given that P(B) has occurred. However, 4+3+2 = 9, which means there is an overlap. By drawing a venn diagram, I got 2 on the outside (neither), 2 in accountant only and 2 in financial planner, then one that is both an accountant and financial planner. If I look at this graph, I can see that out of three accountants, only 1 is an accountant only, so the answer is 1/3. However, I'm not sure how to do it if I want to use the formula. P(A) = 3/8, P(B) = 2/8 ...which would mean P(A|B)= 3/8?

Answer 1

You solved the first two questions correctly.

Let $A$, $F$, and $N$ denote, respectively, the set of qualified candidates who are accountants, financial planners, and neither accountants nor financial planners. We are given \begin{align*} |A \cup F \cup N| & = 8\\ |A| & = 3\\ |F| & = 4\\ |N| & = 2 \end{align*} from which we may conclude that $$|A \cup F| = |A \cup F \cup N| - |N| = 8 - 2 = 6$$
Since $$|A \cup F| = |A| + |F| - |A \cap F|$$ we obtain $$|A \cap F| = |A| + |F| - |A \cup F| = 3 + 4 - 6 = 1$$ Therefore, the probability that a financial planner obtained the job given the fact that an accountant received the job is $$P(F \mid A) = \frac{P(A \cap F)}{P(A)} = \frac{\frac{1}{8}}{\frac{3}{8}} = \frac{1}{8} \cdot \frac{8}{3} = \frac{1}{3}$$ as you found.

For the last question, we want to find $$P(A \mid F^C) = \frac{P(A \cap F^C)}{P(F^C)} = \frac{P(A \setminus F)}{P(F^C)}$$ where $A \setminus F$ is the set of accountants who are not financial planners and $F^C$ is the set of all candidates who are not financial planners.