JEE Mains Maths · Probability

Dice, Digits and Divisibility

Probabilities for dice, coins and coloured draws with unequal chances, and for numbers chosen at random with a condition on their digits or divisibility.

All Probability notes JEE Mains Maths strategy

Why this matters

Sixteen PYQs, fourteen of them multiple choice. Eight throw dice, some with unusual faces or weights, and list the outcomes that give the required total. Eight choose a number at random and ask about divisibility, digits or a gcd — counting problems in probability form. Two ideas cover the page.

Concept 1 of 2: Dice and weighted outcomes

List the pairs (or triples) of faces that give the required result and add their probabilities. For fair dice each pair is

\frac1{36}

; for dice with repeated or weighted faces, multiply the individual face probabilities. Opposite faces of a standard die add to 7, so 'sum 7 with two throws' pairs each face with its opposite.

Definition

Two fair dice: 36 equally likely ordered pairs.
Unequal faces: $P(a,b)=P(a)\,P(b)$ for independent throws.
Sum 7 on two dice: 6 pairs, probability $\frac16$ .
$n$ fair dice: $6^n$ equally likely ordered outcomes.

Independent throws

P(\text{result})=\sum_{\text{favourable }(a,b)}P(a)\,P(b)

Worked example

Two fair dice are thrown. Find the probability that the sum is 9.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 6 April 2023 · Q151Moderate

Example 1 · Probability · Dice, Digits and Divisibility

Three dice are rolled. If the probability of getting different numbers on the three dice is

\frac{p}{q}

, where

p

and

q

are co-prime, then

q - p

is equal to

Ordered pairs

(1,3)

and

(3,1)

are different outcomes on two dice. Listing unordered pairs undercounts every mixed pair by half.

Drill 7 more on dice and weighted outcomes

Concept 2 of 2: Numbers with divisibility or digit properties

A number chosen at random from a range is a counting problem: count the favourable numbers with inclusion–exclusion, residues or digit rules, then divide by the size of the range. For 'coprime to

N

', Euler's function counts them:

\varphi(N)=N\prod\left(1-\frac1p\right)

over the primes dividing

N

Definition

Multiples of $d$ in $1..N$ : $\left\lfloor\frac Nd\right\rfloor$ .
Coprime to $N$ in $1..N$ : $\varphi(N)$ .
Powers mod $m$ repeat: $2^n-2$ is a multiple of 3 exactly when $n$ is odd.
Digits of a random $k$ -digit number: the leading digit has 9 choices, the rest 10.

Euler's function

\varphi(N)=N\prod_{p\mid N}\left(1-\frac1p\right)

Worked example

A number is chosen from 1 to 60. Find the probability that it is coprime to 60.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 29 July 2022 · Q79Moderate

Example 2 · Probability · Dice, Digits and Divisibility

Let

\mathbf{S = \{ 1,2,3,\ldots,2022\}}

. Then the probability, that a randomly chosen number

n

from the set

S

such that

HCF(n,2022) = 1

, is:

The leading digit cannot be 0

In a random

k

-digit number the first digit is 1–9, so it is odd with probability

\frac59

, not

\frac12

. The other digits are 0–9.

Drill 7 more on numbers with divisibility or digit properties

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)

Dice and weighted outcomes
Independent throws
$P(\text{result})=\sum_{\text{favourable }(a,b)}P(a)\,P(b)$
Numbers with divisibility or digit properties
Euler's function
$\varphi(N)=N\prod_{p\mid N}\left(1-\frac1p\right)$

Watch out for (2)

Ordered pairs
→ Dice and weighted outcomes
The leading digit cannot be 0
→ Numbers with divisibility or digit properties

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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