Only two training methods A and B are used for teaching a pa

Only two training methods, A and B, are used for teaching a particular skill. The failure rate for method A is 20% and 10% for method B. Method B, however is expensive and time consuming and it only used 30%. A worker is trained by one of the methods, but fails to learn it. What is the probability that the worker was trained by method A?

Solution

Let F denote the event that he fails to learn it correctly.
By Theorem of Total Probability, we have
P(F) = P(A) * P(F | A) + P(B)P(F | B)
= 0.70 * 0.20 + 0.30 * 0.10
= 0.17
By Bayes’ rule, the probability that he was taught by using method A is
P(A | F) = P(F | A) P(A) / P(F)
= 0.70 * 0.20/0.17
= 0.8235