Step-by-Step Deconstruction

Case Study: Total Probability & Bayes’ Theorem

1. Define Your Events

  • E₁: Customer avails Fixed Rate Loan → P(E₁) = 0.10
  • E₂: Customer avails Floating Rate Loan → P(E₂) = 0.20
  • E₃: Customer avails Variable Rate Loan → P(E₃) = 0.70
  • A: Event that the customer defaults on loan repayment.

Conditional Probabilities:
P(A|E₁) = 0.05 | P(A|E₂) = 0.03 | P(A|E₃) = 0.01

(i) Total Probability of Default

By the Law of Total Probability:

P(A) = P(E₁)P(A|E₁) + P(E₂)P(A|E₂) + P(E₃)P(A|E₃)
P(A) = (0.10 × 0.05) + (0.20 × 0.03) + (0.70 × 0.01)
P(A) = 0.005 + 0.006 + 0.007 = 0.018

(ii) Bayes’ Theorem Application

We need to find P(E₃|A) (Variable rate, given that they defaulted):

P(E₃|A) = [ P(E₃) × P(A|E₃) ] / P(A)
P(E₃|A) = [ 0.70 × 0.01 ] / 0.018
P(E₃|A) = 0.007 / 0.018 = 7/18 (≈ 0.388)

⚠️ Silly Mistake Audit: The Decimal Trap

Most students fail here by misplacing decimals. Remember: 5% is 0.05, NOT 0.5. A mistake at Part (i) will automatically make Part (ii) wrong. Always double-check your total probability sum before moving to Bayes’ Theorem!

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