A system encounters a normal environment with probability 0.85 and a severe environment with probability 0.15. In a normal environment the probability of failure is 0.03, whereas in the severe environment this probability is 0.6
(a)
What is the probability of failure?
(b)
Given that failure has occurred, what is the probability that the environment encountered was severe?

This problem can be approached using **total probability** and **Bayes' Theorem**.

### (a) Probability of failure

The overall probability of failure can be calculated using the law of total probability. The probability of failure depends on the type of environment (normal or severe), and we know:

- Probability of encountering a normal environment, \( P(N) = 0.85 \)
- Probability of failure in a normal environment, \( P(F \mid N) = 0.03 \)
- Probability of encountering a severe environment, \( P(S) = 0.15 \)
- Probability of failure in a severe environment, \( P(F \mid S) = 0.6 \)

The total probability of failure, \( P(F) \), is the sum of the probability of failure in a normal environment and the probability of failure in a severe environment, weighted by the probabilities of encountering each environment. This is calculated as:

\[
P(F) = P(F \mid N) \cdot P(N) + P(F \mid S) \cdot P(S)
\]

Substitute the given values:

\[
P(F) = (0.03 \times 0.85) + (0.6 \times 0.15)
\]

### (b) Probability that the environment was severe given a failure occurred

This part asks for the conditional probability \( P(S \mid F) \), which can be calculated using **Bayes' Theorem**. Bayes' Theorem is given by:

\[
P(S \mid F) = \frac{P(F \mid S) \cdot P(S)}{P(F)}
\]

We've already computed \( P(F) \) in part (a), so we can substitute the known values to find \( P(S \mid F) \).

Now let's compute these values.### (a) The probability of failure is approximately \( P(F) = 0.1155 \), or 11.55%.

### (b) Given that failure has occurred, the probability that the environment encountered was severe is approximately \( P(S \mid F) = 0.779 \), or 77.92%.

Would you like more details or have any further questions? Here are some related questions for further exploration:

1. How does the probability of failure change if the probability of a severe environment increases?
2. What happens to \( P(F) \) if the probability of failure in a severe environment decreases to 0.4?
3. How would the results change if the probabilities for normal and severe environments were equal?
4. How does Bayes' Theorem help in updating probabilities with new information?
5. What is the interpretation of conditional probabilities in risk analysis?

**Tip:** When using Bayes' Theorem, it's essential to know both the likelihood and prior probabilities to update your belief about an event.

A system encounters a normal environment with probability 0.85 and a severe environment with probability 0.15. In a normal environment, the probability of failure is 0.03, whereas in the severe environment, this probability is 0.6. (a) What is the probability of failure? (b) Given that failure has occurred, what is the probability that the environment encountered was severe?

Explore how to calculate the overall probability of failure in a system encountering normal or severe environments, and use Bayes' Theorem to find the conditional probability of a severe environment given failure. This problem involves concepts of total probability and conditional probability, suitable for college-level or advanced high school students.

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