Question

What is the probability that the waste treatment plant is introduced?

Answer 1

Given:

• Let \( A \): First person is appointed, \( P(A) = 0.5 \)
• Let \( B \): Second person is appointed, \( P(B) = 0.6 \)
• Let \( W \): Waste treatment plant is introduced
• \( P(W|A) = 0.7 \), \( P(W|B) = 0.4 \)

Assumption: The appointments are independent, i.e., both persons may be appointed.

Step 1: Apply Total Probability Theorem

\[ P(W) = P(A) \cdot P(W|A) + P(B) \cdot P(W|B) \]

Step 2: Substitute Given Values

\[ P(W) = (0.5)(0.7) + (0.6)(0.4) = 0.35 + 0.24 = 0.59 \]

✅ Final Answer:

\[ \boxed{P(W) = 0.59} \]

Answer 2

Given:

• \( P(A) = 0.5 \), \( P(W|A) = 0.7 \)
• \( P(W) = 0.59 \) (from Total Probability Theorem)

Step 1: Apply Bayes’ Theorem

\[ P(A|W) = \frac{P(A) \cdot P(W|A)}{P(W)} \]

Step 2: Substitute Values

\[ P(A|W) = \frac{0.5 \cdot 0.7}{0.59} = \frac{0.35}{0.59} \]

Step 3: Simplify

\[ P(A|W) = \frac{35}{59} \approx 0.5932 \]

✅ Final Answer:

\[ \boxed{P(A|W) \approx 0.5932} \]