Question

A meeting has 12 employees. Given that 8 of the employees is a woman, find the probability that all the employees are women?

(a) \(\frac{11}{23}\)

(b) \(\frac{12}{35}\)

(c) \(\frac{2}{9}\)

(d) \(\frac{1}{8}\)

I had been asked this question in a national level competition.

This interesting question is from Discrete Probability topic in division Discrete Probability of Discrete Mathematics

Answer 1

The correct answer is (c) \(\frac{2}{9}\)

To explain I would say: Assume that the probability of an employee being a man or woman is (\(\frac{1}{2}\)). By using Bayes’ theorem: let B be the event that the meeting has 3 employees who is a woman and let A be the event that all employees are women. We want to find P(A|B) = \(\frac{P(B|A)*P(A)}{P(B)}\). P(B|A) = 1, P(A) = \(\frac{1}{12}\) and P(B) = \(\frac{8}{12}\). So, P(A|B) = \(\frac{1*\frac{1}{12}}{\frac{8}{12}} = \frac{1}{8}\).