Question

Suppose that there is a chance for a newly constructed house to collapse whether the design is faulty or not. The chance that the design is faulty is 10\%. The chance that the house will collapse if the design is faulty is 95\% and otherwise it is 45\%. It is seen that the house collapsed. What is the probability that the design is faulty?

• A. 0.91
• B. 0.091
• C. 0.19
• D. 0.019

Hint:

Apply Bayes' Theorem for conditional probability.

Answer 1

The Correct Option is C

Step 1: Define events and probabilities.
\( P(F) = 0.10, P(F') = 0.90, P(C|F) = 0.95, P(C|F') = 0.45 \) Step 2: Use Bayes' Theorem: \( P(F|C) = \frac{P(C|F) P(F){P(C)} \).}
Step 3: Calculate \( P(C) \) using total probability.
\( P(C) = P(C|F) P(F) + P(C|F') P(F') = (0.95)(0.10) + (0.45)(0.90) = 0.095 + 0.405 = 0.50 \) Step 4: Calculate \( P(F|C) \).
\( P(F|C) = \frac{(0.95)(0.10)}{0.50} = \frac{0.095}{0.50} = 0.19 \)