JEE Mains Maths · Probability

Counting Favourable Outcomes

Classical probability as a ratio of counts — favourable outcomes over equally likely outcomes — where the counting uses combinations for selections and arrangements for ordered results.

All Probability notes JEE Mains Maths strategy

Why this matters

Twenty-eight PYQs, and 2026 alone has seven. Every one is two counts and a division; the probability is only as right as the sample space is. Nineteen count selections — balls, subsets, pairs, matrices — and nine count ordered results such as words or chosen numbers in order. Two ideas cover the page.

Concept 1 of 2: Selections: counting with combinations

When the order of the chosen items does not matter, count the sample space and the favourable cases both with combinations. For 'at least one', use the complement:

1-P(\text{none})

. Choosing a pair of subsets, or a matrix, counts every entry's choice; keep the sample space and the favourable count in the same terms.

Definition

$P(E)=\frac{n(E)}{n(S)}$ for equally likely outcomes.
Selections: $\binom nr$ in both counts.
At least one: $1-P(\text{none})$ .
Pairs of subsets $(A,B)$ of an $n$ -set: $4^n$ ; with $A\cap B=\varnothing$ : $3^n$ .

Classical probability

P(E)=\frac{n(E)}{n(S)}

Worked example

Two balls are drawn from 5 red and 3 green. Find the probability that both are red.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 22 Jan 2026 Shift 1 · Q54Moderate

Example 1 · Probability · Counting Favourable Outcomes

Two distinct numbers a and b are selected at random from

1,2,3,\ldots\ldots,50

. The probability, that their product ab is divisible by 3 , is

Same sample space in both counts

If the favourable outcomes are counted as ordered pairs, the sample space must be ordered pairs too. Mixing

\binom n2

with

n(n-1)

doubles or halves the answer.

Drill 18 more on selections: counting with combinations

Concept 2 of 2: Ordered outcomes: words, sequences and orders

When order matters, count arrangements. Symmetry is often quickest: in a random arrangement every letter is equally likely to be in any given position, and every relative order of

k

chosen items is equally likely, so 'these two vowels in alphabetical order' has probability

\frac12

. Three numbers chosen at random form an increasing A.P. or G.P. in a countable number of ways — list them by the common difference or ratio.

Definition

Arrangements: $n!$ , or $\frac{n!}{p!\,q!}$ with repeats.
A particular item in a given place: $\frac1n$ by symmetry.
$k$ particular items in one specified order: $\frac{1}{k!}$ .
Increasing A.P. of 3 from $1..n$ : count by the common difference $d$ : $n-2d$ each.

Order by symmetry

P(\text{given relative order of }k\text{ items})=\frac{1}{k!}

Worked example

The letters of PLANE are arranged at random. Find the probability that the vowels come in the order A before E.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 27 June 2022 · Q76Moderate

Example 2 · Probability · Counting Favourable Outcomes

Five numbers

x_{1},x_{2},x_{3},x_{4},x_{5}

are randomly selected from the numbers

1,2,3,\ldots,18

and are arranged in the increasing order

\left( x_{1}<x_{2}<x_{3}<x_{4}<x_{5} \right)

. The probability that

x_{2}= 7

and

x_{4}= 11

is:

Include non-integer ratios

A G.P. of whole numbers can have ratio

\frac32

\frac43

(4,6,9)

(9,12,16)

. Counting only whole-number ratios undercounts.

Drill 8 more on ordered outcomes: words, sequences and orders

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Selections: counting with combinations
Classical probability
$P(E)=\frac{n(E)}{n(S)}$
Ordered outcomes: words, sequences and orders
Order by symmetry
$P(\text{given relative order of }k\text{ items})=\frac{1}{k!}$

Watch out for (2)

Same sample space in both counts
→ Selections: counting with combinations
Include non-integer ratios
→ Ordered outcomes: words, sequences and orders

Test yourself on Probability

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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