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Schooling · Class 12 · Mathematics · Probability · Chapter on a page

CBSE Class 12 · Mathematics · NCERT Mathematics Part-II

Probability: chapter on a page

A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).

About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.

This note's current text has no applied fact-check on record.

Mind map

Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.

Mind map of Probability: Key ideas, Formulas and facts, Worked examples, Watch out forKey ideasConditional probabilityMultiplication ruleIndependent eventsTotal probability theoremBayes' theoremRandom variableMean and varianceFormulas and factsP(A\|B) = P(A ∩ B) / P(B)P(A ∩ B) = P(A) × P(B\|A)P(A ∩ B) = P(A) × P(B)P(A) = Σ P(Bᵢ) × P(A\|Bᵢ)P(Bₖ\|A) = [P(Bₖ) × P(A\|Bₖ)] / P(A)E(X) = Σ xᵢ pᵢVar(X) = E(X²) − [E(X)]²Worked examplesConditional probabilityBayes' theoremBinomial distributionWatch out forUsing P(A) × P(B) when A and B are…Confusing P(A|B) with P(B|A)Forgetting that probabilities in a…Using the binomial formula when…Computing variance as Σ xᵢ² pᵢ…Probability
The mind map as a list
  • Key ideas
    • Conditional probability
    • Multiplication rule
    • Independent events
    • Total probability theorem
    • Bayes' theorem
    • Random variable
    • Mean and variance
  • Formulas and facts
    • P(A\|B) = P(A ∩ B) / P(B)
    • P(A ∩ B) = P(A) × P(B\|A)
    • P(A ∩ B) = P(A) × P(B)
    • P(A) = Σ P(Bᵢ) × P(A\|Bᵢ)
    • P(Bₖ\|A) = [P(Bₖ) × P(A\|Bₖ)] / P(A)
    • E(X) = Σ xᵢ pᵢ
    • Var(X) = E(X²) − [E(X)]²
  • Worked examples
    • Conditional probability
    • Bayes' theorem
    • Binomial distribution
  • Watch out for
    • Using P(A) × P(B) when A and B are…
    • Confusing P(A|B) with P(B|A)
    • Forgetting that probabilities in a…
    • Using the binomial formula when…
    • Computing variance as Σ xᵢ² pᵢ…

What this chapter is about

This chapter builds on the probability ideas you learned in earlier classes and introduces conditional probability, the multiplication rule, independence of events, and Bayes' theorem. These tools let you handle situations where one event affects the chance of another, which is how most real-world uncertainty works.

Key ideas

  • Conditional probability: P(A|B) is the probability of event A given that event B has already occurred. It equals P(A ∩ B) / P(B), provided P(B) > 0.
  • Multiplication rule: For any two events, P(A ∩ B) = P(B) × P(A|B) = P(A) × P(B|A). This lets you find the probability of both events happening together.
  • Independent events: Two events A and B are independent if P(A ∩ B) = P(A) × P(B). Knowing one occurred does not change the probability of the other.
  • Total probability theorem: If B₁, B₂, …, Bₙ form a partition of the sample space, then P(A) = Σ P(Bᵢ) × P(A|Bᵢ).
  • Bayes' theorem: Given prior probabilities of causes B₁, B₂, …, Bₙ and a new observation A, the updated probability is P(Bᵢ|A) = [P(Bᵢ) × P(A|Bᵢ)] / P(A).
  • Random variable: A function that assigns a real number to each outcome of an experiment. Its probability distribution lists every value with its probability.
  • Mean and variance: For a discrete random variable X with values xᵢ and probabilities pᵢ, the mean is E(X) = Σ xᵢ pᵢ and the variance is Var(X) = Σ pᵢ (xᵢ − E(X))² = E(X²) − [E(X)]².
  • Binomial distribution: If an experiment has n independent trials, each with success probability p, the probability of exactly r successes is P(X = r) = C(n, r) × p^r × (1 − p)^(n − r), where C(n, r) is "n choose r".

Formulas and facts to remember

  • Formula: P(A\|B) = P(A ∩ B) / P(B) · Meaning: Conditional probability of A given B.
  • Formula: P(A ∩ B) = P(A) × P(B\|A) · Meaning: Multiplication rule for joint probability.
  • Formula: P(A ∩ B) = P(A) × P(B) · Meaning: Holds only when A and B are independent.
  • Formula: P(A) = Σ P(Bᵢ) × P(A\|Bᵢ) · Meaning: Total probability over a partition.
  • Formula: P(Bₖ\|A) = [P(Bₖ) × P(A\|Bₖ)] / P(A) · Meaning: Bayes' theorem for updating probabilities.
  • Formula: E(X) = Σ xᵢ pᵢ · Meaning: Mean (expectation) of a discrete random variable.
  • Formula: Var(X) = E(X²) − [E(X)]² · Meaning: Variance in shortcut form.
  • Formula: P(X = r) = C(n, r) p^r q^(n − r), q = 1 − p · Meaning: Binomial probability for r successes in n trials.
  • Formula: Mean of binomial = np; Variance = npq · Meaning: Quick results for the binomial distribution.

Common mistakes

  • Using P(A) × P(B) when A and B are not independent → first check independence; if unsure, use the multiplication rule with conditional probability.
  • Confusing P(A|B) with P(B|A) → read the condition after the vertical bar carefully; they are generally different.
  • Forgetting that probabilities in a distribution must sum to 1 → always verify Σ pᵢ = 1 before computing mean or variance.
  • Using the binomial formula when trials are not independent or when the probability changes each trial → binomial requires identical, independent trials.
  • Computing variance as Σ xᵢ² pᵢ without subtracting [E(X)]² → remember Var(X) = E(X²) − [E(X)]².

Quick revision

  • Conditional probability adjusts the sample space to the given condition.
  • Bayes' theorem reverses conditioning: from P(effect|cause) to P(cause|effect).
  • Independent events satisfy P(A ∩ B) = P(A) × P(B).
  • Mean of a random variable is a weighted average of its values, weights being probabilities.
  • Binomial distribution: n trials, constant p, P(X = r) = C(n, r) p^r (1 − p)^(n − r), mean = np, variance = np(1 − p).