Probability

Syllabus mapping

• Unit

  Probability

  Topics

  Probability of an event, Addition and multiplication theorems of probability, Bayes' theorem, Probability distribution of a random variable, Conditional probability, independence and total probability (supporting reasoning)

What this chapter contains and why it matters

• Question

  What is the chapter about?

  Direct answer

  Constructing sample spaces and events, then selecting addition, multiplication, conditional, total-probability or Bayes reasoning.

• Question

  What is the central method choice?

  Direct answer

  Translate wording into set language, check mutual exclusivity vs independence, and validate denominators before conditioning.

• Question

  Where do most mistakes begin?

  Direct answer

  Assuming outcomes are equally likely without justification, and treating mutually exclusive events as independent.

• Question

  What should come before this chapter?

  Direct answer

  Permutations and Combinations for sample-space counting, and Sets and Relations for event modelling.

• Question

  What comes after it?

  Direct answer

  Statistics extends random-variable mean and variance further.

The official JEE documents define content scope. They do not publish chapter weightage, so none is asserted here.

Official JEE syllabus mapping for Probability

Verified against the current JEE Main 2026 syllabus and JEE Advanced 2026 syllabus.

• Concept group

  Explicit topic list

  JEE Main 2026

  Unit 13 explicitly includes probability of an event, addition and multiplication theorems of probability, Bayes' theorem and probability distribution of a random variable. Conditional probability, independence and total probability are not separately named in the Main wording.

  JEE Advanced 2026

  Explicitly includes random experiments, sample spaces, impossible/simple/compound events, addition and multiplication rules, conditional probability, independence, total probability, Bayes' theorem, and probability computation using permutations and combinations.

Sources: JEE Main 2026 syllabus and JEE Advanced 2026 syllabus, both linked in the sources section below.

Before this chapter

• Prerequisite

  Permutations and Combinations

  Probability computation using permutations and combinations requires a settled counting model.

• Prerequisite

  Sets and Relations

  Events are modelled as subsets of a sample space using set operations.

• Parent

  Maths Hub

  Subject hub for the Mathematics chapter set.

Concepts in this chapter

Method selector

• Question signal

  "At least one"

  First model

  Complement is often efficient.

• Question signal

  "A or B"

  First model

  Addition rule, with intersection check.

• Question signal

  "A and B"

  First model

  Multiplication/conditional structure.

• Question signal

  "Given B"

  First model

  Conditional probability, after P(B)>0.

• Question signal

  Cause after observing an outcome

  First model

  Bayes, after partition validation.

• Question signal

  Probability from arrangements/selections

  First model

  Build the P&C sample-space model first.

• Question signal

  Random-variable table

  First model

  Validate probabilities before calculating mean or variance.

Formula sheet

• P(Aᶜ) = 1 - P(A)

  P of A complement equals one minus P of A.

  Probability of the complement of A.

  Use when — "Not A" questions.

  Common trap — Complementing the wrong event.

• P(A∪B) = P(A) + P(B) - P(A∩B)

  P of A union B equals P of A plus P of B minus P of A intersection B.

  Probability of the union of two events.

  Use when — "A or B" questions.

  Common trap — Adding the intersection twice.

• P(A∪B) = P(A) + P(B)

  P of A union B equals P of A plus P of B, valid only when A intersection B is empty.

  Union probability for disjoint events.

  Use when — Disjoint events.

  Common trap — Using this for overlapping events.

• P(A|B) = P(A∩B)/P(B)

  P of A given B equals P of A intersection B divided by P of B, requiring P of B greater than zero.

  Probability of A given B has occurred.

  Use when — "A given B" questions.

  Common trap — Zero denominator.

• P(A∩B) = P(B) P(A|B)

  P of A intersection B equals P of B times P of A given B.

  Intersection probability via conditioning on B.

  Use when — Sequential or intersection events, with P(B)>0.

  Common trap — Assuming independence.

• P(A∩B) = P(A) P(B|A)

  P of A intersection B equals P of A times P of B given A.

  Intersection probability via conditioning on A.

  Use when — Same as multiplication rule, conditioning on A, with P(A)>0.

  Common trap — Zero denominator in the conditional.

• P(A∩B) = P(A) P(B)

  P of A intersection B equals P of A times P of B.

  Definition of independence for events A and B.

  Use when — Testing independence.

  Common trap — Confusing independence with disjointness.

• P(A) = Σ P(A|Bᵢ) P(Bᵢ)

  P of A equals the sum over i of P of A given B sub i times P of B sub i.

  Total probability via a partition of the sample space.

  Bᵢ

  pairwise disjoint, exhaustive partition with P(Bᵢ)>0

  Use when — Combining conditional routes.

  Common trap — Cases not exhaustive or overlapping.

• P(Bⱼ|A) = P(A|Bⱼ) P(Bⱼ) / Σ P(A|Bᵢ) P(Bᵢ)

  P of B sub j given A equals P of A given B sub j times P of B sub j, divided by the sum over i of P of A given B sub i times P of B sub i.

  Reverses the conditioning direction through a partition.

  Bᵢ

  same partition conditions, plus P(A)>0

  Use when — Reverse conditioning.

  Common trap — Missing prior probabilities.

• pᵢ ≥ 0, Σpᵢ = 1

  Each p sub i is nonnegative and the sum of all p sub i equals one.

  Validity condition for a discrete probability distribution.

  Use when — Validate a random-variable table.

  Common trap — Probabilities not summing to 1.

• E[X] = Σxᵢpᵢ

  E of X equals the sum of x sub i times p sub i.

  Expected value of a finite discrete random variable.

  Use when — Advanced random-variable linkage.

  Common trap — Averaging x-values without probabilities.

• Var(X) = Σ(xᵢ-μ)²pᵢ = Σxᵢ²pᵢ - μ²

  Variance of X equals the sum of x sub i minus mu, squared, times p sub i, which must be nonnegative.

  Variance of a finite discrete random variable.

  μ

  E[X]

  Use when — Advanced random-variable linkage.

  Common trap — Negative result from arithmetic error.

Worked examples

Common mistakes and what they actually indicate

• Assuming outcomes are equally likely without justification.

  Needs review

  Why it happens

  Favourable-over-total counting only applies when outcomes are actually equally likely.

  How it is corrected

  Justify the equally-likely assumption before using favourable-over-total counting.

• Confusing union with intersection.

  Knowledge gap

  Why it happens

  "Or" and "and" translate to different set operations with different formulas.

  How it is corrected

  Translate the wording into union or intersection language before selecting a formula.

• Treating mutually exclusive events as independent.

  Knowledge gap

  Why it happens

  If both events have positive probability, mutually exclusive events cannot be independent.

  How it is corrected

  Check both relationships separately rather than assuming one implies the other.

• Cancelling conditional probabilities without a positive denominator.

  Execution error

  Why it happens

  Conditional probability is undefined when the conditioning event has probability zero.

  How it is corrected

  Check that the denominator is positive before using conditional probability.

• Using Bayes without an exhaustive disjoint case structure.

  Decision / selection error

  Why it happens

  Bayes' theorem requires the conditioning cases to form a valid partition.

  How it is corrected

  Verify the partition is pairwise disjoint and exhaustive before applying Bayes' theorem.

• Omitting prior probabilities.

  Recall gap

  Why it happens

  Bayes' theorem requires the prior probability of each partition case.

  How it is corrected

  List all prior probabilities before computing the Bayes numerator and denominator.

• Computing a random-variable mean before validating the distribution.

  Decision / selection error

  Why it happens

  An invalid distribution makes any computed mean meaningless.

  How it is corrected

  Check pᵢ≥0 and Σpᵢ=1 before computing expectation or variance.

• Producing a probability outside [0,1].

  Execution error

  Why it happens

  This signals a modelling or arithmetic error somewhere in the solution.

  How it is corrected

  Validate the final answer lies in [0,1] as a routine check.

Related chapters and pages

• Prerequisite

  Permutations and Combinations

• Related

  Sets and Relations

• Next

  Statistics