JEE Mains Maths · Probability

Random Coefficients and Inequalities

Probabilities where the random outcomes are the coefficients of an equation or the terms of an inequality: count the choices that satisfy the condition.

All Probability notes JEE Mains Maths strategy

Why this matters

Eleven PYQs, nine of them multiple choice. Seven roll or pick the coefficients of a quadratic and ask for real, equal or no real roots, or for positivity for all x; four ask when another inequality holds. The probability is a count once the condition is turned into a statement about the coefficients. Two ideas cover the page.

Concept 1 of 2: Quadratics with random coefficients

Turn the condition into one on the discriminant: real roots need

b^2\ge4ac

, equal roots

b^2=4ac

, distinct real roots

b^2>4ac

, and

ax^2+bx+c>0

for all

x

needs

a>0

and

b^2<4ac

. Then fix

b

and count the pairs

(a,c)

with the required product.

Definition

Real roots: $b^2\ge4ac$ ; equal: $b^2=4ac$ ; none: $b^2<4ac$ .
Positive for all $x$ : $a>0$ and $b^2<4ac$ .
Fix $b$ , count $(a,c)$ with $ac$ on the right side of $\frac{b^2}{4}$ .

Discriminant conditions

ax^2+bx+c:\quad b^2-4ac\ \gtreqless\ 0

Worked example

b

and

c

are each chosen from 1 to 4. Find the probability that

x^2+bx+c=0

has real roots.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 25 July 2022 · Q77Moderate

Example 1 · Probability · Random Coefficients and Inequalities

If the numbers appeared on the two throws of a fair six faced die are

\alpha

and

\beta

, then the probability that

x^{2}+ \alpha x + \beta > 0

, for all

x \in R

, is:

Strict or not

'Real roots' allows

b^2=4ac

; 'two distinct real roots' and 'one root bigger than the other' do not. Check whether the equality cases belong in the count.

Drill 6 more on quadratics with random coefficients

Concept 2 of 2: Other conditions on random outcomes

Solve the inequality in the random quantity first, then count. For a product

x(n-x)\ge k

, solve the quadratic to get a range of

x

. For a sum of dice

N

, find the values of

N

that satisfy the condition, then add the probabilities of those sums.

Definition

Solve for the random variable's range, then count the outcomes in it.
$x(n-x)$ is largest at $x=\frac n2$ .
Sum of two dice: $P(N=k)=\frac{6-|k-7|}{36}$ .

Sum of two dice

P(N=k)=\frac{6-|k-7|}{36},\quad k=2,\dots,12

Worked example

Two dice are thrown. Find the probability that the sum

N

satisfies

N^2>50

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 10 April 2023 · Q72Moderate

Example 2 · Probability · Random Coefficients and Inequalities

Let

N

denotes the sum of the numbers obtained when two dice are rolled. If the probability that

2^{N}< N

! is

\frac{m}{n}

, where

m

and

n

are coprime, then

4m - 3n

is equal to

Integer endpoints

After solving the inequality, check whether the endpoints are integers and whether they are included; a strict inequality drops them.

Drill 3 more on other conditions on random outcomes

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)

Quadratics with random coefficients
Discriminant conditions
$ax^2+bx+c:\quad b^2-4ac\ \gtreqless\ 0$
Other conditions on random outcomes
Sum of two dice
$P(N=k)=\frac{6-|k-7|}{36},\quad k=2,\dots,12$

Watch out for (2)

Strict or not
→ Quadratics with random coefficients
Integer endpoints
→ Other conditions on random outcomes

Test yourself on Probability

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