Step-by-Step Solution

Topic: Total Probability and Bayes’ Theorem

1. Define Events and Probabilities

First, find the total number of seeds: 10 (Brinjal) + 12 (Cabbage) + 8 (Radish) = 30 seeds.

Let us define the events:

  • E₁: Selecting a Brinjal seed. P(E₁) = 10/30
  • E₂: Selecting a Cabbage seed. P(E₂) = 12/30
  • E₃: Selecting a Radish seed. P(E₃) = 8/30
  • A: The selected seed germinates.

Given germination probabilities:

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

(I) TOTAL PROBABILITY OF GERMINATION

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) = (10/30)(0.25) + (12/30)(0.35) + (8/30)(0.40)
P(A) = (1/3)(0.25) + (2/5)(0.35) + (4/15)(0.40)
P(A) = 0.0833 + 0.14 + 0.1067

Total Probability P(A) = 0.33 (or 33/100)

(II) PROBABILITY IT’S CABBAGE, GIVEN IT GERMINATES

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

P(E₂|A) = [ P(E₂) ⋅ P(A|E₂) ] / P(A)

Substitute the values:

P(E₂|A) = [ (12/30) ⋅ 0.35 ] / 0.33
P(E₂|A) = [ 0.4 ⋅ 0.35 ] / 0.33
P(E₂|A) = 0.14 / 0.33 = 14 / 33

Final Probability: P(E₂|A) = 14/33 (≈ 0.424)

⚠️ Silly Mistake Audit: The “Initial Probability” Trap

A very common mistake is to calculate the initial probabilities (like P(E₁)) incorrectly. Remember to always divide the number of seeds of a specific type by the TOTAL number of seeds (30), not just by 100. Also, ensure you convert percentages to decimals correctly (e.g., 25% = 0.25) before calculating.

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