Step-by-Step Deconstruction

Topic: Bayes’ Theorem & Conditional Probability

1. Identify the Events

• E₁: Student has misconception → P(E₁) = 0.40
• E₂: Student does NOT have misconception → P(E₂) = 0.60
• B: Student answers “Bijoy” in the test.

Conditional Probabilities:
P(B|E₁) = 0.80 | P(B|E₂) = 1 – 0.90 = 0.10

(i) P(Not Misconception AND Bijoy)

P(E₂ ∩ B) = P(E₂) × P(B|E₂) = 0.60 × 0.10 = 0.06

(ii) Total Probability P(B)

P(B) = P(E₁)P(B|E₁) + P(E₂)P(B|E₂)
P(B) = (0.40 × 0.80) + (0.60 × 0.10) = 0.32 + 0.06 = 0.38

(iii) (a) Bayes’ Theorem for Misconception

P(E₁|B) = [P(E₁)P(B|E₁)] / P(B) = 0.32 / 0.38 = 16/19 (≈0.842)

(iii) (b) Bayes’ Theorem for No Misconception

P(E₂|B) = 1 – P(E₁|B) = 1 – 16/19 = 3/19 (≈0.158)

⚠️ Silly Mistake Audit: The “NOT” Translation

Pay close attention to the phrasing! The problem states 90% did NOT answer Bijoy. Many students use 0.90 directly in Part (ii). Always verify your conditional values: If 90% of a group doesn’t do something, then 10% (0.10) must do it.