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3.Bayes’ Theorem
Class 12 • Mathematics • NCERT
Bayes’ Theorem
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This lesson explains Bayes’ Theorem, which helps in finding conditional probabilities by reversing conditions, as prescribed in NCERT Class 12.
Lesson Objectives
- Understand the statement of Bayes’ Theorem.
- Apply Bayes’ Theorem to calculate probabilities.
- Use prior and posterior probabilities.
- Solve NCERT board-level problems.
1. Need for Bayes’ Theorem
Bayes’ Theorem is used to find the probability of an event when some related event has already occurred.
It reverses the condition in conditional probability.
[ Image Placeholder: Tree Diagram Showing Reversal of Conditions ]
2. Statement of Bayes’ Theorem (NCERT)
If A₁, A₂, …, Aₙ are mutually exclusive and exhaustive events and B is any event such that P(B) ≠ 0, then:
P(Aᵢ | B) = [ P(Aᵢ) · P(B | Aᵢ) ] / Σ [ P(Aⱼ) · P(B | Aⱼ) ]
[ Image Placeholder: Mutually Exclusive and Exhaustive Events ]
3. Prior and Posterior Probabilities
The probabilities P(A₁), P(A₂), … are called prior probabilities, and P(Aᵢ | B) is called the posterior probability.
Prior → Before experiment
Posterior → After observing event B
Posterior → After observing event B
[ Image Placeholder: Prior vs Posterior Probability ]
4. Conditions for Applying Bayes’ Theorem
Bayes’ Theorem can be applied only if certain conditions are satisfied.
• Events A₁, A₂, … must be mutually exclusive
• Events must be exhaustive
• P(B) ≠ 0
• Events must be exhaustive
• P(B) ≠ 0
[ Image Placeholder: Conditions Illustration ]
5. NCERT Example
A bag contains two coins C₁ and C₂.
C₁ is a fair coin and C₂ is a biased coin with probability of head = 3/4.
A coin is chosen at random and tossed. It shows head.
Find the probability that the coin chosen was C₂.
C₁ is a fair coin and C₂ is a biased coin with probability of head = 3/4.
A coin is chosen at random and tossed. It shows head.
Find the probability that the coin chosen was C₂.
[ Image Placeholder: Coin Selection Tree Diagram ]
6. Important NCERT Notes
• Bayes’ Theorem uses conditional probability
• Widely used in probability problems
• Prior and posterior probabilities must be clearly identified
• Very important board-exam topic
• Widely used in probability problems
• Prior and posterior probabilities must be clearly identified
• Very important board-exam topic
Practice Questions (NCERT)
- State Bayes’ Theorem.
- What are prior probabilities?
- What are posterior probabilities?
- Why should events be exhaustive?
- Can Bayes’ Theorem be applied if P(B)=0?
- What type of events are A₁, A₂, …?
- Which theorem helps reverse conditional probability?
- Is Bayes’ Theorem part of NCERT Class 12 syllabus?
- Is it important for board exams?
- Is Bayes’ Theorem based on conditional probability?
✅ Show Answer Key
- Statement using mutually exclusive and exhaustive events
- Probabilities before experiment
- Probabilities after observing event
- To cover entire sample space
- No
- Mutually exclusive and exhaustive
- Bayes’ Theorem
- Yes
- Yes
- Yes
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