Step-by-Step Solution

Topic: Total Probability & Bayes’ Theorem

1. Define Probabilities

Total Population = 1000.

Probabilities of Health Categories:

• • P(A₁) = 700/1000 = 0.7 (Good Health)
• • P(A₂) = 200/1000 = 0.2 (Average Health)
• • P(A₃) = 100/1000 = 0.1 (Poor Health)

Conditional Probabilities of Contracting Disease (Let E be the event):

• • P(E|A₁) = 25% = 0.25
• • P(E|A₂) = 35% = 0.35
• • P(E|A₃) = 50% = 0.50

(I) PROBABILITY OF CONTRACTING DISEASE

Using the Law of Total Probability:

Substitute the values:

P(E) = (0.7)(0.25) + (0.2)(0.35) + (0.1)(0.50)

P(E) = 0.175 + 0.070 + 0.050

(II) PROBABILITY OF A₂ GIVEN NO DISEASE

Let E’ be the event of not contracting the disease.

First, find P(E’):

P(E’) = 1 – P(E) = 1 – 0.295 = 0.705

Next, find the probability of not contracting disease given A₂:

P(E’|A₂) = 1 – P(E|A₂) = 1 – 0.35 = 0.65

Using Bayes’ Theorem:

Substitute the values:

P(A₂|E’) = (0.2 × 0.65) / 0.705

P(A₂|E’) = 0.130 / 0.705

Multiply numerator and denominator by 1000:

P(A₂|E’) = 130 / 705

⚠️ Silly Mistake Audit: The “Not” Trap

A common mistake in Part (ii) is using the probability of contracting the disease (0.295) as the denominator, or using 0.35 as the conditional probability. The question asks for the case where the person has NOT contracted the disease. You must calculate the complement probabilities \( 1 – P(E) \) and \( 1 – P(E|A_2) \) before applying Bayes’ theorem.