Core Syllabus Module

Probability

Systematic mastery engine. Select a core micro-topic target below or simulate custom examination constraints.

Exam Simulator

Custom Chapter Mock Test

Generate random timed arrays filtered strictly by Probability weightage mappings.

Launch Simulator

Micro-Topic Strategy Matrix

Chapter Elite-20 Challenge Zone

Target Advanced

JEE Main 2023 (24 January Shift 2)

LEVELJEE Advanced

The urns A, B and C contain 4 red, 6 black; 5 red, 5 black and $λ$ red, 4 black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn C is 0.4 then the square of the length of the side of the largest equilateral triangle, inscribed in the parabola $y^{2} = λ x$ with one vertex at the vertex of the parabola is

JEE Advanced 2023

LEVELJEE Advanced

Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is $\frac{1}{3}$ , then the probability that the experiment stops with head is.

(A)

\frac{1}{3}

(B)

\frac{5}{21}

(C)

\frac{4}{21}

(D)

\frac{2}{7}

JEE Advanced 2019

LEVELJEE Advanced

Let $∣ X ∣$ denote the number of elements in set $X$ . Let $S = {1, 2, 3, 4, 5, 6}$ be a sample space, where each element is equally likely to occur. If $A$ and $B$ are independent events associated with $S$ , then the number of ordered pairs $(A, B)$ such that $1 \leq ∣ B ∣ < ∣ A ∣$ , equals

JEE Advanced 2000

LEVELJEE Advanced

A coin has probability $p$ of showing head when tossed. It is tossed $n$ times. Let $p_{n}$ denote the probability that no two (or more) consecutive heads occur. Prove that $p_{1} = 1, p_{2} = 1 - p^{2}$ and $p_{n} = (1 - p) p_{n - 1} + p (1 - p) p_{n - 2}$ for all $n \geq 3$ . Prove by induction on $n$ , that $p_{n} = A α^{n} + B β^{n}$ for all $n \geq 1$ , where $α$ and $β$ are the roots of quadratic equation $x^{2} - (1 - p) x - p (1 - p) = 0$ and $A = \frac{p ^{2} + β - 1}{α β - α ^{2}}, B = \frac{p ^{2} + α - 1}{α β - β ^{2}}$ .

JEE Main 2021 (March)

LEVELJEE Advanced

Let A denote the event that a 6-digit integer formed by 0,1,2,3,4,5,6 without repetitions,be divisible by 3. Then probability of event A is equal to :

(A)

9/50

(B)

4/9

(C)

3/7

(D)

11/27

JEE Main 2025 April

LEVELJEE Advanced

All five letter words are made using all the letters A, B, C, D, E and arranged as in an English dictionary with serial numbers. Let the word at serial number $n$ be denoted by $W_{n}$ . Let the probability of choosing the word $W_{n}$ satisfy $P (W_{n}) = 2 P (W_{n - 1}), n > 1$ . If $P (C D B E A) = \frac{2 ^{α}}{2 ^{β} - 1}$ , $α, β \in N$ , then $α + β$ is equal to : _______

JEE Advanced 2024

LEVELJEE Advanced

Let $X$ be a random variable, and let $P (X = x)$ denote the probability that $X$ takes the values $x$ . Suppose that the points $(x, P (X = x))$ , $x = 0, 1, 2, 3, 4$ , lie on a fixed straight line in the $x y$ -plane, and $P (X = x) = 0$ for all $x \in R - {0, 1, 2, 3, 4}$ . If the mean of $X$ is $\frac{5}{2}$ and the variance of $X$ is $α$ , then the value of $24 α$ is ________.

JEE Advanced 2006

LEVELJEE Advanced

Comprehension Passage

There are

n

urns, each of these contain

n + 1

balls. The

i^{t h}

urn contains

i

white balls and

(n + 1 - i)

red balls. Let

u_{i}

be the event of selecting

i^{t h}

urn,

i = 1, 2, 3, \dots, n

and

w

the event of getting a white ball.

Question 1:

If $P (u_{i}) \propto i$ , where $i = 1, 2, 3, \dots, n$ , then $lim_{n \to \infty} P (w) =$

If $P (u_{i}) = c$ , (a constant) then $P (u_{n} / w) =$

(A)

\frac{2}{n + 1}

(B)

\frac{1}{n + 1}

(C)

\frac{n}{n + 1}

(D)

\frac{1}{2}

Question 3:

Let $P (u_{i}) = \frac{1}{n}$ , if $n$ is even and $E$ denotes the event of choosing even numbered urn, then the value of $P (w / E)$ is

(A)

\frac{n + 2}{2 n + 1}

(B)

\frac{n + 2}{2 ( n + 1 )}

(C)

\frac{n}{n + 1}

(D)

\frac{1}{n + 1}

JEE Advanced 1986

LEVELJEE Advanced

A lot contains 20 articles. The probability that the lot contains exactly 2 defective articles is 0.4 and the probability that the lot contains exactly 3 defective articles is 0.6. Articles are drawn from the lot at random one by one without replacement and are tested till all defective articles are found. What is the probability that the testing procedure ends at the twelfth testing?

JEE Main 2021 (22 July Shift 1)

LEVELJEE Advanced

Four dice are thrown simultaneously and the numbers shown on these dice are recorded in $2 \times 2$ matrices. The probability that such formed matrices have all different entries and are non-singular, is :

(A)

45/162

(B)

23/81

(C)

22/81

(D)

43/162

JEE Advanced 2020

LEVELJEE Advanced

Let $C_{1}$ and $C_{2}$ be two biased coins such that the probabilities of getting head in a single toss are $\frac{2}{3}$ and $\frac{1}{3}$ , respectively. Suppose $α$ is the number of heads that appear when $C_{1}$ is tossed twice, independently, and suppose $β$ is the number of heads that appear when $C_{2}$ is tossed twice, independently. Then probability that the roots of the quadratic polynomial $x^2 - \alpha x + eta$ are real and equal, is

(A)

\frac{40}{81}

(B)

\frac{20}{81}

(C)

\frac{1}{2}

(D)

\frac{1}{4}

JEE Advanced 2015

LEVELJEE Advanced

Comprehension Passage

Let

n_{1}

and

n_{2}

be the number of red and black balls, respectively, in box I. Let

n_{3}

and

n_{4}

be the number of red and black balls, respectively, in box II.

Question 1:

One of the two boxes, box I and box II, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box II is 1/3, then the correct option(s) with the possible values of $n_{1}, n_{2}, n_{3}$ and $n_{4}$ is(are)

* Multiple Correct Options

(A)

n_{1} = 3, n_{2} = 3, n_{3} = 5, n_{4} = 15

(B)

n_{1} = 3, n_{2} = 6, n_{3} = 10, n_{4} = 50

(C)

n_{1} = 8, n_{2} = 6, n_{3} = 5, n_{4} = 20

(D)

n_{1} = 6, n_{2} = 12, n_{3} = 5, n_{4} = 20

Question 2:

A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is 1/3, then the correct option(s) with the possible values of $n_{1}$ and $n_{2}$ is(are)

* Multiple Correct Options

(A)

n_{1} = 4

and

n_{2} = 6

(B)

n_{1} = 2

and

n_{2} = 3

(C)

n_{1} = 10

and

n_{2} = 20

(D)

n_{1} = 3

and

n_{2} = 6

JEE Advanced 2021

LEVELJEE Main

Let $E, F$ and $G$ be three events having probabilities $P (E) = \frac{1}{8}, P (F) = \frac{1}{6}$ and $P (G) = \frac{1}{4}$ , and let $P (E \cap F \cap G) = \frac{1}{10}$ . For any event $H$ , if $H^{c}$ denotes its complement, then which of the following statements is(are) TRUE ?

* Multiple Correct Options

(A)

P (E \cap F \cap G^{c}) \leq \frac{1}{40}

(B)

P (E^{c} \cap F \cap G) \leq \frac{1}{15}

(C)

P (E \cup F \cup G) \leq \frac{13}{24}

(D)

P (E^{c} \cap F^{c} \cap G^{c}) \leq \frac{5}{12}

JEE Main 2022 (28 June Shift 2)

LEVELJEE Main

The probability that a randomly chosen one-one function from the set ${a, b, c, d}$ to the set ${1, 2, 3, 4, 5}$ satisfies $f (a) + 2 f (b) - f (c) = f (d)$ is :

(A)

\frac{1}{24}

(B)

\frac{1}{40}

(C)

\frac{1}{30}

(D)

\frac{1}{20}

JEE Advanced 2022

LEVELJEE Main

Suppose that Box-I contains 8 red, 3 blue and 5 green balls, Box-II contains 24 red, 9 blue and 15 green balls, Box-III contains 1 blue, 12 green and 3 yellow balls, Box-IV contains 10 green, 16 orange and 6 white balls. A ball is chosen randomly from Box-I; call this ball $b$ . If $b$ is red then a ball is chosen randomly from Box-II, if $b$ is blue then a ball is chosen randomly from Box-III, and if $b$ is green then a ball is chosen randomly from Box-IV. The conditional probability of the event 'one of the chosen balls is white' given that the event 'at least one of the chosen balls is green' has happened, is equal to

(A)

\frac{15}{256}

(B)

\frac{3}{16}

(C)

\frac{5}{52}

(D)

\frac{1}{8}

JEE Advanced 2022

LEVELJEE Main

Two players, $P_{1}$ and $P_{2}$ , play a game against each other. In every round of the game, each player rolls a fair die once, where the six faces of the die have six distinct numbers. Let $x$ and $y$ denote the readings on the die rolled by $P_{1}$ and $P_{2}$ , respectively. If $x < y$ , then $P_{1}$ scores 5 points and $P_{2}$ scores 0 point. If $x = y$ , then each player scores 2 points. If $x > y$ , then $P_{1}$ scores 0 point and $P_{2}$ scores 5 points. Let $X_{i}$ and $Y_{i}$ be the total scores of $P_{1}$ and $P_{2}$ , respectively, after playing the $i^{t h}$ round.

List-I

(P)

Probability of

(X_{2} \geq Y_{2})

(Q)

Probability of

(X_{2} > Y_{2})

(R)

Probability of

(X_{3} = Y_{3})

(S)

Probability of

(X_{3} > Y_{3})

List-II

(1)

3/8

(2)

11/16

(3)

5/16

(4)

355/864

(5)

77/432

JEE Main 2023 (29 January Shift 1)

LEVELJEE Main

Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is

(A)

\frac{5}{24}

(B)

\frac{2}{15}

(C)

\frac{1}{6}

(D)

\frac{5}{36}

JEE Advanced 2011

LEVELJEE Main

Comprehension Passage

Let

U_{1}

and

U_{2}

be two urns such that

U_{1}

contains 3 white and 2 red balls, and

U_{2}

contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from

U_{1}

and put into

U_{2}

. However, if tail appears then 2 balls are drawn at random from

U_{1}

and put into

U_{2}

. Now 1 ball is drawn at random from

U_{2}

Question 1:

The probability of the drawn ball from $U_{2}$ being white is

Given that the drawn ball from $U_{2}$ is white, the probability that head appeared on the coin is

Comprehension Passage

A box

B_{1}

contains 1 white ball, 3 red balls and 2 black balls. Another box

B_{2}

contains 2 white balls, 3 red balls and 4 black balls. A third box

B_{3}

contains 3 white balls, 4 red balls and 5 black balls.

Question 1:

If 1 ball is drawn from each of the boxes $B_{1}, B_{2}$ and $B_{3}$ , the probability that all 3 drawn balls are of the same colour is

If 2 balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these 2 balls are drawn from box $B_{2}$ is

In how many ways three girls and nine boys can be seated in two vans, each having numbered seats, 3 in the front and 4 at the back? How many seating arrangements are possible if 3 girls should sit together in a back row on adjacent seats? Now, if all the seating arrangements are equally likely, what is the probability of 3 girls sitting together in a back row on adjacent seats?

Micro-Topic Strategy Matrix

Random Experiments and Sample Space

Classical Definition of Probability

Addition and Multiplication Theorems

Conditional Probability

Total Probability Theorem

Bayes' Theorem

Random Variables and Probability Distributions

Binomial Distribution

Chapter Elite-20 Challenge Zone

Let A denote the event that a 6-digit integer formed by 0,1,2,3,4,5,6 without repetitions,be divisible by 3. Then probability of event A is equal to :

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If P(ui​)∝i, where i=1,2,3,…,n, then limn→∞​P(w)=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If P(ui​)=c, (a constant) then P(un​/w)=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let P(ui​)=n1​, if n is even and E denotes the event of choosing even numbered urn, then the value of P(w/E) is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The probability that a randomly chosen one-one function from the set {a,b,c,d} to the set {1,2,3,4,5} satisfies f(a)+2f(b)−f(c)=f(d) is :

List-I

List-II

Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The probability of the drawn ball from U2​ being white is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Given that the drawn ball from U2​ is white, the probability that head appeared on the coin is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If 1 ball is drawn from each of the boxes B1​,B2​ and B3​, the probability that all 3 drawn balls are of the same colour is

If $P (u_{i}) \propto i$ , where $i = 1, 2, 3, \dots, n$ , then $lim_{n \to \infty} P (w) =$

If $P (u_{i}) = c$ , (a constant) then $P (u_{n} / w) =$

Let $P (u_{i}) = \frac{1}{n}$ , if $n$ is even and $E$ denotes the event of choosing even numbered urn, then the value of $P (w / E)$ is

The probability that a randomly chosen one-one function from the set ${a, b, c, d}$ to the set ${1, 2, 3, 4, 5}$ satisfies $f (a) + 2 f (b) - f (c) = f (d)$ is :

The probability of the drawn ball from $U_{2}$ being white is

Given that the drawn ball from $U_{2}$ is white, the probability that head appeared on the coin is

If 1 ball is drawn from each of the boxes $B_{1}, B_{2}$ and $B_{3}$ , the probability that all 3 drawn balls are of the same colour is