Step-by-Step Solution

Topic: Total Probability & Bayes’ Theorem

Probability Tree Diagram

Branch 1: Person A (0.5) → Plant (0.7).
Branch 2: Person B (0.6) → Plant (0.4).

1. Define Events & Probabilities

• • Let E₁: First person is appointed. P(E₁) = 0.5
• • Let E₂: Second person is appointed. P(E₂) = 0.6
• • Let A: Waste treatment plant is introduced.

Conditional Probabilities:

• • P(A|E₁) = 0.7 (Plant introduced by Person 1)
• • P(A|E₂) = 0.4 (Plant introduced by Person 2)

(I) TOTAL PROBABILITY OF PLANT INTRODUCTION

Using the Law of Total Probability:

Substitute the values:

P(A) = (0.5 × 0.7) + (0.6 × 0.4)

P(A) = 0.35 + 0.24

(II) PROBABILITY OF PERSON 1 (Given Plant Introduced)

We need to find P(E₁|A) using Bayes’ Theorem:

Substitute the values calculated in Part (I):

Numerator = P(E₁)P(A|E₁) = 0.5 × 0.7 = 0.35

Denominator = P(A) = 0.59

Therefore:

P(E₁|A) = 0.35 / 0.59