Model Answer
0 min readIntroduction
In risk assessment and decision-making, understanding the probability of an event given certain conditions is paramount. This is particularly relevant in fields like engineering and construction, where failures can have significant consequences. The problem presented is a classic application of Bayes' Theorem, which allows us to update our belief about an event based on new evidence. Bayes' Theorem is a mathematical formula for determining conditional probability – the probability of an event occurring given that another event has already occurred. In this case, we need to determine the probability that a house collapse is due to a design fault, given that the house has already collapsed.
Understanding the Problem
Let's define the events:
- D: The design of the house is faulty.
- C: The house collapses.
We are given the following probabilities:
- P(D) = 0.1 (Probability that the design is faulty)
- P(C|D) = 0.95 (Probability that the house collapses given the design is faulty)
- P(C|¬D) = 0.45 (Probability that the house collapses given the design is NOT faulty)
We need to find P(D|C), the probability that the design is faulty given that the house has collapsed.
Bayes' Theorem
Bayes' Theorem is stated as:
P(D|C) = [P(C|D) * P(D)] / P(C)
Where P(C) is the probability of the house collapsing, which can be calculated using the law of total probability:
P(C) = P(C|D) * P(D) + P(C|¬D) * P(¬D)
Calculating P(C)
First, we need to find P(¬D), the probability that the design is not faulty:
P(¬D) = 1 - P(D) = 1 - 0.1 = 0.9
Now, we can calculate P(C):
P(C) = (0.95 * 0.1) + (0.45 * 0.9) = 0.095 + 0.405 = 0.5
Calculating P(D|C)
Now we can apply Bayes' Theorem:
P(D|C) = (0.95 * 0.1) / 0.5 = 0.095 / 0.5 = 0.19
Conclusion
Therefore, the probability that the house collapsed due to a fault in the design is 0.19 or 19%.
Conclusion
In conclusion, using Bayes' Theorem, we determined that even though the probability of a faulty design is only 10%, the probability that a collapse is *due* to a faulty design, given that a collapse has occurred, is 19%. This highlights the importance of considering prior probabilities and conditional probabilities when assessing risk and causality. This approach is crucial in various fields, including quality control, medical diagnosis, and financial risk management.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.