Step-by-Step Solution

Topic: Total Probability & Bayes’ Theorem

1. Define Events & Probabilities

Let E₁, E₂, E₃ be the events of choosing a phone from Company A, B, and C respectively.

  • • P(E₁) = 25% = 0.25
  • • P(E₂) = 35% = 0.35
  • • P(E₃) = 40% = 0.40

Let D be the event that the phone is defective.

  • • P(D|E₁) = 5% = 0.05
  • • P(D|E₂) = 4% = 0.04
  • • P(D|E₃) = 2% = 0.02

(I) TOTAL PROBABILITY OF DEFECT

Using the Law of Total Probability:

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

Substitute the values:

P(D) = (0.25)(0.05) + (0.35)(0.04) + (0.40)(0.02)

P(D) = 0.0125 + 0.0140 + 0.0080

Total Probability P(D) = 0.0345

(II) PROBABILITY OF COMPANY B (Given Defective)

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

P(E₂|D) = [ P(E₂) × P(D|E₂) ] / P(D)

Substitute the values calculated in Part (I):

P(E₂|D) = (0.35 × 0.04) / 0.0345

P(E₂|D) = 0.0140 / 0.0345

To simplify, multiply numerator and denominator by 10,000:

P(E₂|D) = 140 / 345

Divide both by 5:

Final Probability = 28/69 (≈ 0.4058)

⚠️ Silly Mistake Audit: Decimal Conversion

A frequent error in these problems is converting single-digit percentages. Remember that 5% is 0.05, NOT 0.5. Using 0.5 would mean 50% of the phones are defective, which would drastically change your final answer. Always double-check your decimals!

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