Example 1 · level 1
For events A and B, P(B) = 21/25 and P(A ∩ B) = 4/5. Find P(A|B).
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- P(A|B) = (4/5) ÷ (21/25) = 20/21 = 20/21.
Answer: 20/21
CLASS 12 MATHS · PROBABILITY
Conditional probability and Bayes’ theorem
P(A|B) = P(A ∩ B)/P(B). Bayes: P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ) / Σ P(Eⱼ)P(A|Eⱼ), where the Eⱼ partition the sample space.
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Worked examples
Example 2 · level 2
Two dice are thrown. Given that the sum is 5, find the probability that at least one die shows 4.
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- Outcomes with sum 5: 4.
- Of these, 2 contain a 4.
- P = 2/4 = 1/2.
Answer: 1/2
Example 3 · level 3
A company makes LED bulbs in Pune, Kochi and Surat, producing 20%, 60% and 20% of its output. Of these, 1%, 1% and 2% respectively are defective. A bulb chosen at random is defective. Find the probability that it was made in Kochi.
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- P(defective) ∝ 20 × 1 + 60 × 1 + 20 × 2 = 120.
- P(Kochi | defective) = 60/120 = 1/2.
Answer: 1/2