Class 12 Mathematics Chapter 13: Probability
CBSE 2026–27 | Detailed NCERT-aligned free resource
This chapter develops conditional probability, multiplication rules, independent events, total probability and Bayes’ theorem. The focus is on translating situations carefully into events and conditional probabilities.
1. Conditional Probability
For events A and B with P(B)>0, P(A|B)=P(A∩B)/P(B). It measures the probability of A when B is already known to have occurred.
“Probability of A given B”
Worked Example
A card is drawn from a standard deck. Given that it is a face card, what is the probability that it is a king? There are 12 face cards and 4 kings, so P(King|Face)=4/12=1/3.
2. Multiplication Theorem
P(A∩B)=P(A)P(B|A)=P(B)P(A|B).
For three events, P(A∩B∩C)=P(A)P(B|A)P(C|A∩B).
3. Independent Events
A and B are independent when P(A∩B)=P(A)P(B). Equivalently, when probabilities are defined, P(A|B)=P(A) and P(B|A)=P(B).
4. Total Probability
If B₁,B₂,…,Bₙ form a partition of the sample space, then P(A)=ΣP(Bᵢ)P(A|Bᵢ).
B₂ ──┼──→ A
B₃ ──┘
P(A)=ΣP(Bᵢ)P(A|Bᵢ)
5. Bayes’ Theorem
P(Bᵢ|A)=P(Bᵢ)P(A|Bᵢ) / ΣP(Bⱼ)P(A|Bⱼ).
Worked Example: Defective Items
Machine A produces 60% of items and has a defect rate of 2%. Machine B produces 40% and has a defect rate of 5%. If a randomly selected item is defective, the probability it came from B is [0.40×0.05]/[0.60×0.02+0.40×0.05] = 0.02/0.032 = 0.625.
6. Independent vs Mutually Exclusive
| Concept | Meaning |
|---|---|
| Mutually exclusive | A∩B=∅; they cannot occur together |
| Independent | Occurrence of one does not change the probability of the other |
Non-trivial mutually exclusive events are not independent because P(A∩B)=0 while P(A)P(B)>0.
7. How to Identify the Correct Formula
- “Given that” usually signals conditional probability.
- “Both A and B” signals an intersection.
- “Either/or” requires attention to overlap.
- Multiple sources or groups followed by an observed outcome often indicates total probability or Bayes’ theorem.
8. Common Mistakes
- Reversing P(A|B) and P(B|A).
- Assuming events are independent without evidence.
- Using Bayes’ theorem without calculating the total probability of the observed event.
- Confusing mutually exclusive with independent events.
9. Practice Set
- Calculate a conditional probability from a deck of cards.
- Use the multiplication theorem for dependent events.
- Test whether two events are independent.
- Use total probability for a problem involving multiple factories.
- Apply Bayes’ theorem when an item is known to have a particular property.
- Explain why two non-zero mutually exclusive events cannot be independent.