# Probability Notes, Class 12 Mathematics | SwaVid

Probability, NCERT Class 12 Mathematics: notes with key ideas, formulas, common mistakes and 3 quick checks with worked answers.

Canonical: https://www.swavid.com/cbse/class-12/maths/chapter/probability

Source: https://www.swavid.com/cbse/class-12/maths/chapter/probability

# Probability

## Formulas and definitions

## Key ideas

## Common mistakes

## Quick checks

## Keep going

## Questions parents ask

## Class 12 Mathematics on SwaVid

### Conditional probability

### Properties of conditional probability

### Multiplication rule of probability

### Independent events

### Partition of a sample space

### Theorem of total probability

### Bayes' theorem

### Related chapter

### Related chapter

### All chapters

SwaVid › CBSE › Class 12 › Mathematics › Probability

Notes written by SwaVid from the NCERT textbook.

Class 12 Mathematics, Chapter 13. Key ideas, formulas and common mistakes.

CBSE does not print marks for a single chapter.

Short answer. This chapter covers conditional probability, the multiplication rule, independent events, and Bayes' theorem. Students learn how prior knowledge changes event likelihoods and how to reverse probabilities using given evidence.

Source: NCERT Mathematics Part II (Reprint 2026-27) , Chapter 13, Probability

The conditional probability of an event E given that event F has already occurred is denoted by P(E|F). It is calculated as P(E ∩ F) / P(F), provided P(F) ≠ 0.

For a sample space S and events A and B, P(S|F) = P(F|F) = 1. Also, P((A ∪ B)|F) = P(A|F) + P(B|F) - P((A ∩ B)|F).

For two events E and F, P(E ∩ F) = P(E) × P(F|E), provided P(E) ≠ 0. For three events, P(E ∩ F ∩ G) = P(E) × P(F|E) × P(G|(E ∩ F)).

Two events E and F are independent if the occurrence of one does not affect the probability of the other. Mathematically, E and F are independent if and only if P(E ∩ F) = P(E) × P(F).

A set of events E₁, E₂, ..., Eₙ forms a partition of S if they are pairwise disjoint, their union is S, and each event has a non-zero probability.

If E₁, E₂, ..., Eₙ form a partition of S, then for any event A, P(A) = ∑ P(Eᵢ) × P(A|Eᵢ) summed from i = 1 to n.

If E₁, E₂, ..., Eₙ form a partition of S and A is an event of non-zero probability, the posterior probability P(Eᵢ|A) is (P(Eᵢ) × P(A|Eᵢ)) / (∑ P(Eⱼ) × P(A|Eⱼ)).

Try each question first. Then tap it to see the answer.

P(A ∩ B) = P(A) × P(B|A) = 0.8 × 0.4 = 0.32

P(A|B) = P(A ∩ B) / P(B) = 0.32 / 0.5 = 0.64

P(A ∩ B) = 0.32, P(A|B) = 0.64

P(E) = 3/6 = 1/2 and P(F) = 2/6 = 1/3

E ∩ F = {3}, so P(E ∩ F) = 1/6

P(E) × P(F) = (1/2) × (1/3) = 1/6

Since P(E ∩ F) = P(E) × P(F), the events are independent

Yes, E and F are independent

P(E₁) = P(E₂) = 1/2, where E₁ and E₂ represent selecting Bag I and Bag II

P(Red|E₁) = 3/7 and P(Red|E₂) = 5/11

P(Red) = P(E₁)P(Red|E₁) + P(E₂)P(Red|E₂)

P(Red) = (1/2)(3/7) + (1/2)(5/11) = 3/14 + 5/22 = (33 + 35) / 154 = 68/154 = 34/77

34/77

Relations and Functions .

Linear Programming .

The 13 chapters of Class 12 Mathematics .

CBSE does not give marks to one chapter. It prints marks for units or parts, and the chapter list page shows them all.

No. They are written by SwaVid from the NCERT textbook. The CBSE curriculum is the reference for what is assessed.

This is the last chapter in the NCERT books for Class 12 Mathematics.

SwaVid is an AI tutor for Indian students in Classes 6 to 12. It finds the earlier ideas a child missed, then teaches today’s chapter. Start free to see how it works.

- Ch 13 NCERT book
- 80 marks, theory paper
- 3 quick checks
- Confusing mutually exclusive events with independent events . Mutually exclusive events satisfy P(E ∩ F) = 0, whereas independent events satisfy P(E ∩ F) = P(E) × P(F). Two events with positive probabilities cannot be both mutually exclusive and independent.
- Swapping the condition and the event in conditional probability notation . Remember that in P(A|B), B is the given event that has already occurred, and A is the event whose probability is being found. The denominator is always P(B).
- Assuming events are independent without checking their intersection probability . Do not assume independence just because events seem unrelated. Always verify whether P(E ∩ F) equals P(E) × P(F).

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