Step-by-Step Deconstruction

Topic: Total Probability & Bayes’ Theorem

Step 1: Define Events & Probabilities

Let E₁, E₂, E₃ be the events that a car is manufactured by Amber, Bonzi, and Comet respectively. Let A be the event that the car is electric.

• P(E₁) = 0.60 | P(E₂) = 0.30 | P(E₃) = 0.10
• P(A|E₁) = 0.20 | P(A|E₂) = 0.10 | P(A|E₃) = 0.05

(i) (a) Total Probability of Electric Car [P(A)]

P(A) = P(E₁)P(A|E₁) + P(E₂)P(A|E₂) + P(E₃)P(A|E₃)

P(A) = (0.60)(0.20) + (0.30)(0.10) + (0.10)(0.05)

P(A) = 0.12 + 0.03 + 0.005 = 0.155

(i) (b) Total Probability of Petrol Car

P(Petrol) = 1 – P(Electric) = 1 – 0.155 = 0.845

(ii) Bayes’ Theorem for Comet [P(E₃|A)]

P(E₃|A) = [ P(E₃)P(A|E₃) ] / P(A)

P(E₃|A) = (0.10 × 0.05) / 0.155 = 0.005 / 0.155 = 5 / 155 = 1 / 31

(iii) Probability for Amber or Bonzi

We need P(E₁|A) + P(E₂|A) OR 1 – P(E₃|A).

Result = 1 – (1/31) = 30 / 31

⚠️ Silly Mistake Audit: The “OR” Logic

In part (iii), don’t restart the whole Bayes’ calculation for both Amber and Bonzi. Since these are mutually exclusive producers, the probability of “Amber OR Bonzi” is simply 1 minus the probability of Comet. Using this shortcut saves time and reduces calculation errors!