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We are tasked with solving part (b) of the problem: 

**Part (b)**: "If dangerous pollution is detected in the plant discharge, what is the probability that process A was operating?"

This is a problem of conditional probability. To solve this, we need to use **Bayes' Theorem**, which is formulated as:

\[
P(A | \text{pollution detected}) = \frac{P(\text{pollution detected} | A) \cdot P(A)}{P(\text{pollution detected})}
\]

Let's break it down:

- \(P(A)\) is the probability that process A is operating. The relative likelihoods of the processes (A, B, C, D) are given as 2:4:3:1. The total sum of the weights is \(2 + 4 + 3 + 1 = 10\). Hence, the probability of process A operating is:
  \[
  P(A) = \frac{2}{10} = 0.2
  \]

- \(P(\text{pollution detected} | A)\) is the probability that pollution is detected given that process A is operating. This is provided directly in the problem as 40%, or:
  \[
  P(\text{pollution detected} | A) = 0.40
  \]

- \(P(\text{pollution detected})\) is the total probability that pollution is detected, regardless of which process is running. We calculate this by using the law of total probability:
  \[
  P(\text{pollution detected}) = P(\text{pollution detected} | A)P(A) + P(\text{pollution detected} | B)P(B) + P(\text{pollution detected} | C)P(C) + P(\text{pollution detected} | D)P(D)
  \]
  From the problem, the probabilities of detecting pollution for each process are:
  - \(P(\text{pollution detected} | B) = 0.05\),
  - \(P(\text{pollution detected} | C) = 0.30\),
  - \(P(\text{pollution detected} | D) = 0.10\).

  The probabilities of each process operating are based on their relative likelihoods:
  - \(P(B) = \frac{4}{10} = 0.4\),
  - \(P(C) = \frac{3}{10} = 0.3\),
  - \(P(D) = \frac{1}{10} = 0.1\).

Now, I will calculate the total probability \(P(\text{pollution detected})\).The total probability that pollution is detected, \( P(\text{pollution detected}) \), is 0.2.

Now we can apply Bayes' Theorem to calculate \( P(A | \text{pollution detected}) \):

\[
P(A | \text{pollution detected}) = \frac{P(\text{pollution detected} | A) \cdot P(A)}{P(\text{pollution detected})}
\]

Substituting the known values:

\[
P(A | \text{pollution detected}) = \frac{0.40 \times 0.2}{0.2}
\]

I will now calculate this.The probability that process A was operating, given that dangerous pollution was detected, is 0.4 or 40%.

Do you need further clarification or details on this part? 

Here are some additional related questions:
1. How would the result change if the probabilities of pollution for each process were different?
2. What if process A's probability of operating was reduced, how would that affect the outcome?
3. How can Bayes' Theorem be applied in real-world scenarios beyond this problem?
4. How would we handle this problem if more processes were involved?
5. What is the significance of conditional probability in decision-making?

**Tip:** Always double-check the probabilities and dependencies when applying Bayes' Theorem, as small errors in values can affect the outcome greatly.

If dangerous pollution is detected in the plant discharge, what is the probability that process A was operating?

Learn how to apply conditional probability and Bayes' Theorem to calculate the likelihood that process A was operating when dangerous pollution was detected. This solution explains key concepts such as Bayes' Theorem and the Law of Total Probability.

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