AdvanceReview
…Checking

Math notesAlgebraProbability

Independent topic page

Probability & Statistics

Probability full marks come from modelling: define events → partition/tree → apply definition/Bayes/counting → keep exact fractions. Statistics (mean/variance) uses the same clean table habit.

Portion AlgebraFocus High; Bayes + counting joins appear every few papersTarget 60–90sPortion share 37.2%Trend Falling share
Official syllabus includes total probability, Bayes, and P&C-based probability, plus mean and variance ideas. Never start with a formula name — start with the sample-space story. Probability marks follow the model; formulae only finish a correct tree or table.

How Advanced asks this

  • Bayes / total probability story stems
  • Conditional probability with clear P(A|B)
  • Independence checks in MC
  • Counting-based probability
  • Mean and variance of a discrete RV

Traps that leak marks

  • Assuming independence without check
  • Confusing ME with independence
  • Forgetting priors in Bayes
  • Variance = E(X²) only
  • Treating mutually exclusive as independent (or the reverse)
  • Shipping E(X²) as variance
  • Different counting models in num vs den of a probability

01. Most wrong answers are model errors, not arithmetic. A tree forces mutually exclusive routes that cover the experiment.

02. Independence ≠ mutually exclusive. Write the test P(A∩B)=P(A)P(B) when MC claims independence.

03. Counting probability: build the P&C model sentence first (see P&C notes), then divide favourable by total.

04. For discrete random variables: make the value table, compute E(X) and E(X²), then Var=E(X²)−[E(X)]².

05. Advanced lane: Bayes is almost always a tree or a partition table. Draw the first split (cause / type / bag) before writing any formula — then Bayes is arithmetic on those leaves.

06. Counting-probability joins: freeze the same model sentence for numerator and denominator. Changing ‘order matters’ mid-stem is how multi-correct clusters die.

07. Random variable stems: build the value→probability table first; E(X), E(X²), Var follow mechanically. Never jump to Var without E(X²).

Official · 2026

Syllabus for this topic

Official syllabus points for Probability & Statistics only — not the full Algebra portion map. Portion hub: Algebra · Algebra analysis.

Syllabus on this page is topic-scoped: Probability & Statistics. The parent portion (Algebra) also covers other topics — use the portion hub for the full map.

Must cover (this topic)

  • Bayes / total probability
  • Mean & variance of a random variable

Gaps / watch-outs

  • Statistics (often under-revised vs probability)

Source: https://jeeadv.ac.in/documents/jee-advanced-2026-syllabus.pdf

Unit 01

Probability & Statistics

high

Exam focusConditional probability, Bayes, independence, and random-variable mean/variance.

Syllabus points

  • Sample space; impossible/simple/compound events
  • Addition and multiplication rules; conditional probability; independence
  • Total probability; Bayes’ theorem; P&C-based probability
  • Mean, median, mode; mean deviation; SD and variance (grouped/ungrouped)
  • Same mean, different variance comparisons; random variable mean & variance

Do / check

  • Draw tree / partition before Bayes
  • Distinguish independence from mutually exclusive

Full Mathematics syllabus →

Cross-topic stems

Mixed joins for this topic

Advanced rarely asks a chapter in isolation. These are tagged multi-idea Math stems that touch Probability & Statistics — practise the knowledge bridge, not only the single-chapter drill.

No tagged mixed stem matched this topic’s keywords yet. Still revise the joins listed under Trends, and scan the mixed joins board for Calculus / Coordinate / Algebra bridges.

All mixed joins →

Why methods matter

Common methods that change your score

Probability marks leak on wrong models, not wrong arithmetic. These methods make you draw the story first — so Bayes and counting stop feeling like lucky guesses.

01

Tree first

If the tree is right, Bayes is arithmetic. If you skip the tree, marks become luck.

Do thisDraw 2–3 branch tree with priors and conditionals before naming Bayes.

Why it works

Makes the denominator of Bayes automatic.

See it once

Two factories 60%/40%, defect 2%/5% → P(A|def)=(0.012)/(0.032)=3/8.

02

Model sentence for counting-P

Keeps numerator and denominator in the same counting world — the #1 P&C-probability leak.

Do thisWrite P&C model, then favourable/total.

Why it works

Stops mismatched numerator/denominator counting rules.

See it once

If order mattered in total, it must matter in favourable (or both use combinations).

03

Independence vs ME audit

One confusion rule saves multi-correct clusters every year.

Do thisFor MC, test independence and mutually exclusive as separate T/F.

Why it works

These two are the most confused pair in options.

See it once

ME ⇒ P(A∩B)=0. If P(A),P(B)>0 they cannot be independent.

04

RV table

Makes variance mechanical so you never ship E(X²) as Var by mistake.

Do thisList x, p(x), xp, x²p in columns.

Why it works

Prevents missing E(X²) for variance.

See it once

Var without E(X²)−[E(X)]² is the #1 statistics leak.

Processed examples

Sample questions · full working

Read the impact line first. Then notice which family/model was chosen. The solution is written the way your rough sheet should look in the hall.

01

standardFamily · Bayes / two-source story

Sample question

Factory A makes 60% items (2% defective), B makes 40% (5% defective). An item is defective. Find P(from A).

This is the Bayes tree prototype. If you skip the tree, you skip the denominator.

Model chosenTree: priors × defect rates → normalise.

Processed solution

  1. P(A)=0.6, P(B)=0.4; P(D|A)=0.02, P(D|B)=0.05.
  2. P(D)=0.02·0.6 + 0.05·0.4 = 0.032.
  3. P(A|D)=0.012/0.032 = 3/8.

Answer3/8

Takeaway: Draw the tree before writing Bayes.

02

bankFamily · Independence vs mutually exclusive MC

Sample question

Events A,B have P(A)=0.3, P(B)=0.4, P(A∩B)=0. If a statement says ‘A and B are independent’, is it true?

ME vs independence confusion is a yearly MC trap.

Model chosenME ⇒ ∩=0; Ind needs ∩=P·P.

Processed solution

  1. P(A∩B)=0 ⇒ mutually exclusive.
  2. For independence need 0.3·0.4=0.12, but ∩=0 ≠ 0.12.
  3. So not independent (since both have positive probability).

AnswerFalse — not independent

Takeaway: Two separate T/F tests, never one fuzzy word.

03

twistFamily · Bayes / two-source story

Sample question

Boxes B₁, B₂, B₃ chosen with probabilities proportional to 1 : 2 : 3; red fractions 1/2, 1/3, 1/6. Given red, find P(B₂).

Unequal priors kill ‘average the fractions’ shortcuts — the tree must carry weights.

Model chosenNormalise priors → multiply by red rates → Bayes.

Processed solution

  1. Priors: P(B₁)=1/6, P(B₂)=2/6, P(B₃)=3/6.
  2. Branch weights for red: (1/6)(1/2)=1/12, (2/6)(1/3)=1/9, (3/6)(1/6)=1/12.
  3. P(red) = 1/12 + 1/9 + 1/12 = 5/18.
  4. P(B₂|red) = (1/9) / (5/18) = 2/5.

Answer2/5

Takeaway: Multiply prior × likelihood on every branch before normalising.

Reusable models

Common question families

Advanced stems cluster. Name the family in 10–15s, write the fast model, then run the same path. Do not invent a new plan each time.

01

Bayes / two-source story

Looks likeTwo factories/urns/tests; ask P(source | result)

Fast modelDraw a 2–3 branch tree with priors × likelihoods before naming Bayes.

Same solve path

  1. Priors P(Bᵢ).
  2. Likelihoods P(A|Bᵢ).
  3. Total P(A)=∑…
  4. Posterior ∝ likelihood×prior; normalise.

Rough skeleton

  • Tree
  • P(A)
  • Posterior

Why faster

The tree is the answer engine; formula recall without tree causes missing denominators.

Refuse this

Do not jump to Bayes formula without writing the partition.

02

Conditional definition

Looks like‘Given that’, P(A|B), restricted sample space

Fast modelRewrite as P(A∩B)/P(B); redefine the sample space to the given event.

Same solve path

  1. Identify A and B in words.
  2. Compute intersection and P(B).
  3. Divide; keep exact fractions.

Rough skeleton

  • A,B
  • P(A∩B)
  • P(B)
  • Ratio

Why faster

Stops using absolute probabilities where conditional is required.

Refuse this

Do not use P(A) when the stem says ‘given B’.

03

Counting-based probability

Looks likeCards/committees/digits with probability language

Fast modelWrite the P&C model sentence first (order? replacement?), then favourable/total with the SAME model.

Same solve path

  1. Model sentence for total.
  2. Same model for favourable.
  3. P = fav/total.
  4. Overcount check.

Rough skeleton

  • Model
  • n(S)
  • n(A)
  • P

Why faster

Mismatched nPr/nCr between num and den is the #1 leak — model lock prevents it.

Refuse this

Do not mix combinations in numerator with permutations in denominator.

04

Independence vs mutually exclusive MC

Looks likeWhich statements true about independent / ME events

Fast modelTwo separate T/F lines: ME ⇒ P∩=0; Ind ⇒ P∩=P·P — never conflate.

Same solve path

  1. Test ME claim.
  2. Test independence claim.
  3. If both P>0, ME ⇒ not independent.
  4. Tick options independently.

Rough skeleton

  • ME?
  • Ind?
  • Options T/F

Why faster

One confusion rule kills half the distractors.

Refuse this

Do not treat ‘cannot occur together’ as independence.

05

Discrete mean–variance

Looks likeRandom variable table; mean / variance

Fast modelColumns x, p, xp, x²p → E then Var=E(X²)−[E(X)]².

Same solve path

  1. Check ∑p=1.
  2. E(X), E(X²).
  3. Var formula.
  4. Var≥0 sanity.

Rough skeleton

  • ∑p
  • E
  • E(X²)
  • Var

Why faster

Table habit prevents the classic ‘Var=E(X²)’ mistake.

Refuse this

Do not skip E(X²) or forget to subtract [E(X)]².

Hall timing

60–90s process

  1. 0–15s
    Define events in one line each. Note conditional vs absolute ask.

    Underline ‘given that’. If factories/urns/diseases appear, prepare Bayes tree.

  2. 15–35s
    Draw partition/tree or equally likely sample space. Write priors and likelihoods.

    For equally likely outcomes, state n(S) carefully (order/replacement).

  3. 35–75s
    Apply definition / Bayes / counting; keep exact fractions.

    Bayes: posterior ∝ likelihood × prior; normalise by total probability once.

  4. 75–90s
    Check 0≤P≤1 and that partition probabilities sum to 1.

    For variance, confirm you subtracted [E(X)]². Reject options >1 or negative.

After the model is locked

Quick formula bank

Classical probability

P(A)=n(A)/n(S) (equally likely)

Fair coins/dice/cards with clear sample space

Addition

P(A∪B)=P(A)+P(B)−P(A∩B)

Union asks

Conditional

P(A|B)=P(A∩B)/P(B)

‘Given that’ wording

Independence

P(A∩B)=P(A)P(B)

Independence claims

Total probability

P(A)=∑ P(A|Bᵢ)P(Bᵢ)

Partition {Bᵢ}

Bayes

P(Bᵢ|A)=P(A|Bᵢ)P(Bᵢ)/∑P(A|Bⱼ)P(Bⱼ)

Reverse probability / which source given outcome

Complement

P(A′)=1−P(A)

At least one / none shortcuts

Mean & variance

E(X)=∑x p(x); Var(X)=E(X²)−[E(X)]²

Discrete RV statistics

Binomial (when applicable)

Mean np; variance npq

Independent Bernoulli trials count

Mental bank

  • P(A|B)=P(A∩B)/P(B)
  • Bayes = reverse the tree
  • Independent: P(A∩B)=P(A)P(B)
  • ME: P(A∩B)=0
  • Var=E(X²)−[E(X)]²
  • 0 ≤ P ≤ 1 always
  • P(A|B)=P(A∩B)/P(B) — write ∩ before dividing
  • Independent ⇒ P(A∩B)=P(A)P(B); mutually exclusive ⇒ P(A∩B)=0
  • Var(X)=E(X²)−[E(X)]²
  • Total probability: partition the sample space, then weight

Make it automatic

Practice · checklist

Daily

  • 1 Bayes tree + 1 conditional definition stem.
  • 1 counting-probability with written model sentence.
  • 1 mean/variance table (alt days).

Weekly

  • 1 MC set on independence vs ME.
  • 1 mixed Probability+P&C set.
  • Error log: missing Bayes denominators.

100% checklist

  • Events defined in words before formulae.
  • Tree/partition drawn for Bayes.
  • Independence tested, not assumed.
  • Variance uses both E(X²) and [E(X)]².
  • ≥90% on last timed Probability set.
  • Tree or partition drawn for every Bayes / total-P stem.
  • Independence vs ME checked in words before a product or zero.
  • RV table complete before any expectation formula.
Algebra hubAll Math notesSyllabusSubject analysisMixed joins