Question

70% of the total employees of a factory are men. Among the employees of that factory, 30% of men and 15% of women are technical assistants. If an employee chosen at random is found to be a technical assistant, then the probability that this employee is a man is:

• A. \(\frac{9}{23}\)
• B. \(\frac{3}{17}\)
• C. \(\frac{14}{17}\)
• D. \(\frac{14}{23}\)

Hint:

Use total probability and Bayes theorem for mixed conditional probabilities.

Answer 1

The Correct Option is C

Step 1: Define events
Let \(M\) be the event employee is a man, \(W\) is employee is a woman.
\[ P(M) = 0.7, \quad P(W) = 0.3 \] Step 2: Given
\[ P(T | M) = 0.3, \quad P(T | W) = 0.15 \] Step 3: Total probability of technical assistant
\[ P(T) = P(T | M) P(M) + P(T | W) P(W) = 0.3 \times 0.7 + 0.15 \times 0.3 = 0.21 + 0.045 = 0.255 \] Step 4: Find \(P(M | T)\) using Bayes theorem
\[ P(M | T) = \frac{P(T | M) P(M)}{P(T)} = \frac{0.21}{0.255} = \frac{14}{17} \]