Step-by-Step Solution

Topic: Total Probability & Bayes’ Theorem

1. Define Events & Probabilities

Let A be the event that the airplane reaches late.

Let E1, E2, and E3 be the events of Severe, Moderate, and Light turbulence respectively.

Probabilities of Turbulence (Equal):

P(E1) = P(E2) = P(E3) = 1/3

Conditional Probabilities (Late given Turbulence):

  • • P(A|E1) = 55% = 0.55
  • • P(A|E2) = 37% = 0.37
  • • P(A|E3) = 17% = 0.17

(I) TOTAL PROBABILITY OF BEING LATE

Using the Law of Total Probability:

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

Substitute the values:

P(A) = (1/3)(0.55) + (1/3)(0.37) + (1/3)(0.17)

P(A) = (1/3) [ 0.55 + 0.37 + 0.17 ]

P(A) = (1/3) [ 1.09 ]

Total Probability P(A) = 1.09 / 3 ≈ 0.3633

(II) PROBABILITY OF MODERATE TURBULENCE (Given Late)

We need to find P(E2|A) using Bayes’ Theorem:

P(E2|A) = [ P(E2) × P(A|E2) ] / P(A)

Numerator: P(E2)P(A|E2) = (1/3) × 0.37

Denominator: P(A) = (1/3) × 1.09

Since (1/3) is common in both numerator and denominator, it cancels out:

P(E2|A) = 0.37 / 1.09

Multiply by 100 to remove decimals:

Probability = 37/109 (≈ 0.339)

⚠️ Silly Mistake Audit: Missing the ‘1/3’

Don’t forget the Prior Probability!
A common mistake in Total Probability questions is to simply add the percentages (0.55 + 0.37 + 0.17) without multiplying by the probability of the turbulence occurring (1/3). This would give you a probability > 1, which is impossible. Always multiply P(A|E) by P(E).

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