Probability (Class 12 Maths): Conditional Probability & Bayes’ Theorem Solved
The formulas that matter and a step-by-step Bayes’ theorem example — a common and scoring Class 12 board question.
Quick idea: Conditional probability measures the chance of A given B has happened. Bayes’ theorem “reverses” the condition — finding a cause given an observed effect.
The core formulas
P(A|B) = P(A ∩ B) / P(B)
Bayes: P(Eᵢ|A) = P(Eᵢ)·P(A|Eᵢ) / Σ P(Eⱼ)·P(A|Eⱼ)
Worked example — Bayes’ theorem
Q. Two bags: Bag I has 3 red, 2 black; Bag II has 1 red, 4 black. A bag is chosen at random and a red ball is drawn. What is the probability it came from Bag I?
P(I) = P(II) = 1/2. P(red|I) = 3/5, P(red|II) = 1/5.
P(I|red) = (1/2×3/5) / (1/2×3/5 + 1/2×1/5) = (3/10)/(4/10) = 3/4
How to approach these questions
- List the events and their prior probabilities.
- Write the conditional probabilities given each event.
- Plug into Bayes and simplify carefully.
Common mistakes
- Swapping P(A|B) with P(B|A).
- Forgetting to sum all cases in the denominator.
- Not simplifying fractions at the end.
FAQs
Is Bayes’ theorem important for boards?
Yes — it is a recurring long-answer question in Class 12. Confirm current weightage from the syllabus.
Does this help for entrance exams?
Yes — conditional probability and Bayes appear in JEE and other exams too.
Need help with Class 12 Maths?
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