Knowledge organisers

Probability: CBSE Class 12 Maths knowledge organiser

Everything to know about probability on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Conditional probability: P(A | B) = P(A ∩ B)/P(B), the probability of A given B.
Independent events: P(A ∩ B) = P(A)P(B).
Bayes' theorem: Finds the probability of a cause given that an event has happened.

Key formulas

Conditional	\(P(E\mid F)=\frac{P(E\cap F)}{P(F)},\) \(P(F)\ne0\)
Properties	\(P(S\mid F)=1,\) \(P(E'\mid F)=1-P(E\mid F)\)
Multiplication rule	\(P(E\cap F)=P(E)P(F\mid E)=P(F)P(E\mid F)\)
Independent	\(P(E\cap F)=P(E)P(F)\)
Union	\(P(E\cup F)=P(E)+P(F)-P(E\cap F)\)
Total probability (partition \(E_1,\dots,E_n\))	\(P(A)=\sum_{j=1}^nP(E_j)P(A\mid E_j)\)
Bayes' theorem	\(P(E_i\mid A)=\frac{P(E_i)P(A\mid E_i)}{\sum_{j=1}^nP(E_j)P(A\mid E_j)}\)
Random variable	\(p_i\ge0,\) \(\sum p_i=1,\) \(E(X)=\mu=\sum x_ip_i\)

Worked example

A card is drawn from a pack of 52. Given it is a face card (J, Q, K), the probability that it is a king is \(\dfrac{4}{12} = \dfrac13\).

Common mistakes

Dividing by the wrong event: in \(P(A \mid B)\) the given event \(B\) goes in the denominator.
Assuming independence without justification, or confusing it with mutually exclusive.
In Bayes’ questions, forgetting a branch in the denominator or not defining \(E_1, E_2, \dots\) and \(A\).
Probability distributions whose probabilities do not add to \(1\) — always check.

You should be able to…

Compute conditional probabilities from counts and formulas, use the multiplication theorem and find k in a probability distribution.
Test independence, apply total probability and Bayes’ theorem, and find the mean of a random variable at board 2- and 3-mark level.
Write complete 5-mark Bayes and random-variable answers (events defined, tree/table, formula, conclusion) and handle probability case studies.
Handle proofs about conditional probability and independence, sequential evidence in Bayes, and distributions built from without-replacement experiments.

The printable sheet

Probability knowledge organiser for CBSE Class 12 Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Probability knowledge organiser (CBSE Class 12 Maths), A4. Download the PDF.

Revise it next

Other CBSE Class 12 Maths topics: Relations and Functions · Inverse Trigonometric Functions · Matrices · Determinants · Continuity and Differentiability · Application of Derivatives · Integrals · Application of Integrals · Differential Equations · Vector Algebra · Three Dimensional Geometry · Linear Programming · All CBSE Class 12 Maths organisers