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Playbook

Probability

Counting in disguise, then Bayes' theorem and the binomial distribution. Set up the sample space once and count both parts the same way.

Questions in the bank
149
q/paper in 2025–26
1.11
Numeric answer
19%
Notes pages
7

Tier: Core

When you’ll see it

The chance of an event: balls from bags, dice, a number picked at random, a 'given that', or a distribution with a mean and a variance.

How this chapter is tested

Much of Probability on JEE Mains is counting in disguise. Counting favourable outcomes and the dice, digits and divisibility page are two counts and a division, and the counts come from Permutations and Combinations. Get the sample space right first; the division is the easy part.

The rest is formula work. Among the formula pages, Bayes' theorem comes up most often: total probability over the bags, then one branch divided by the total. The binomial distribution and random variables ask for a probability, a mean or a variance. Trials repeated until a success sum a geometric series, and quadratics with random coefficients turn a probability into a discriminant condition.

Time goes on setting up, not on arithmetic. Most wrong answers count ordered outcomes in one place and unordered ones in another, or use a denominator that has lost a ball. When a numeric-answer question gives the probability as m/n in lowest terms and asks for m + n, reduce the fraction fully before adding.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Counting favourable outcomes

    Selections by combinations, ordered results by permutations — both counts over the same sample space.

  • Dice, digits and divisibility

    List ordered outcomes on dice, including odd or weighted faces; count numbers with a digit or divisibility property.

  • Random coefficients

    Turn 'real roots' into b² ≥ 4ac, or another inequality into a condition on the outcomes, then count.

  • Addition, conditional and independent events

    P(A ∪ B) = P(A) + P(B) − P(A ∩ B), P(A | B) = P(A ∩ B)/P(B), independence as a product, repeated trials as a geometric series.

  • Total probability and Bayes' theorem

    Sum over every bag or machine with its prior, then divide the branch asked for by that total.

  • Binomial distribution

    P(X = r) = C(n, r) pʳ qⁿ⁻ʳ; mean np and variance npq, so q = variance ÷ mean.

  • Random variables

    Make the probabilities add to 1, then E(X) = Σ x P(x) and Var(X) = E(X²) − (E(X))².

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • Ordered and unordered mixed

    (1, 3) and (3, 1) are two outcomes on two dice. Count the favourable cases and the sample space the same way, or the answer doubles or halves.

  • The bag after a transfer

    The receiving bag holds one more ball; its old total in the denominator is wrong. Unequal priors must stay in every product.

  • Squaring the wrong thing

    E(X²) = Σ x² P(x). Squaring the probabilities, or using (E(X))² in place of E(X²), gives a wrong variance.

  • The variance is the smaller

    Since q < 1, npq < np. When the mean and variance come from one quadratic, the mean is the larger root.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Probability notes

Drill every Probability question

149 questions from the bank, across 7 subtopics.

Drill the 149 questions

Drill one subtopic at a time

The 7 subtopics, in teaching order.

Related playbooks

Often paired with this one — the technique or the trap overlaps. Drill these next.