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JEE • MATHEMATICS • PROBABILITY

Probability practice · Hard

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Practice format and chapter quiz

These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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1 Hard

1 of 23

Analyzing +4 | -1

A bag contains four tickets marked with numbers 112, 121, 211, 222. One ticket is drawn at random from the bag. Let

{E_i}\left( {i = 1,\,2,\,3} \right)

denote the event that ith digit on the ticket is 2. Then, which one of the following is incorrect?

View answer and explanation

Correct option: D

{E_1}

{E_2}

and

{E_3}

are independent

The correct answer is that

{E_1}

{E_2}

and

{E_3}

are not independent because

P(E_1 \cap E_2 \cap E_3) \ne P(E_1)P(E_2)P(E_3)

2 Hard

2 of 23

Application +4 | -1

A bag contains some white and some
black balls, all combinations of balls being equally likely. The
total number of balls in the bag is 10. If three balls are drawn
at random without replacement and all of them are found to
be black, the probabili ty that the bag contains 1 white and
9 black balls is

View answer and explanation

Correct option: A

14 / 55

Option A (14/55) is correct because it is the conditional probability calculated using Bayes' theorem, given that three black balls were drawn.

3 Hard

3 of 23

Understanding +4 | -1

A

and

B

are independent events such that

P\left( A \right) > 0,P\left( B \right) > 0,

then

View answer and explanation

Correct option: D

P\left( {A/B} \right) + P\left( {\bar A/B} \right) = 1

Option D (

P(A/B) + P(\bar A/B) = 1

) is correct because

P(A/B) + P(\bar A/B)

represents the sum of conditional probabilities of an event and its complement given B, which always equals 1.

4 Hard

4 of 23

Application +4 | -1

In an assembly of 4 persons the probability that at least 2 of them have the same birthday, is

View answer and explanation

Correct option: D

0.016

Option D (0.016) is correct because it approximates the probability that at least two people out of four share the same birthday.

5 Hard

5 of 23

Application +4 | -1

Let

A

be a set containing

n

elements. A subset

P

of the set

A

is chosen at random. The set

A

is reconstructed by replacing the elements of

P

, and another subset

Q

A

is chosen at random. The probability that

P \cap Q

contains exactly

m(m < n)

elements, is

View answer and explanation

Correct option: C

\frac{{ }^n C_m \times 3^{n-m}}{4^n}

Option C (

\frac{{^n C_m \times 3^{{n-m}}}}{{4^n}}

) is correct because it calculates the probability of P and Q having exactly m elements in common by considering the combinations and dividing by the total possible outcomes.

6 Hard

6 of 23

Application +4 | -1

Three dice are thrown. The probability of getting a sum which is a perfect square, is

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the calculated probability of getting a sum that is a perfect square is

\frac{17}{108}

, which is not among the given options.

7 Hard

7 of 23

Application +4 | -1

Two dice are thrown

n

times in succession. The probability of obtaining a double six at least once is

View answer and explanation

Correct option: B

1 - {\left( {\frac{{35}}{{36}}} \right)^n}

Option B (

1 - {(\frac{{35}}{{36}})^n}

) is correct because it represents the probability of obtaining at least one double six in n throws, calculated as 1 minus the probability of not obtaining any double sixes.

8 Hard

8 of 23

Application +4 | -1

The probability that the roots of the equation

{x^2} + nx + \frac{1}{2} + \frac{n}{2} = 0

are real where

n \in N

such that

n \le 5,

View answer and explanation

Correct option: D

04-May

Option D (04-May) is correct because it calculates the probability of the roots being real based on the possible values of n and the discriminant condition.

9 Hard

9 of 23

Analyzing +4 | -1

Out of

3n

consecutive natural numbers, 3 natural numbers are chosen at random without replacement. The probability that the sum of the chosen numbers is divisible by 3, is

View answer and explanation

Correct option: C

\frac{{\left( {3{n^2} - 3n + 2} \right)}}{{\left( {3n - 1} \right)\left( {3n - 2} \right)}}

Option B (

\frac{{\left( {3{n^2} - 3n + 2} \right)}}{{2\left( {3n - 1} \right)\left( {3n - 2} \right)}}

) is correct because it represents the probability that the sum of three randomly chosen numbers from 3n consecutive natural numbers is divisible by 3.

10 Hard

10 of 23

Application +4 | -1

A cricket club has 15 members, of whom only 5 can bowl. If the names of 15 members are put into a box and 11 are drawn at random, then the probability of obtaining an eleven containing at least 3 bowlers is

View answer and explanation

Correct option: D

Dec-13

Option D (12/13) is correct because it calculates the probability of obtaining an eleven containing at least 3 bowlers by considering all possible combinations of bowlers and other players.

11 Hard

11 of 23

Analyzing +4 | -1

Three dice are thrown. The probability of getting a sum which is a perfect square, is

View answer and explanation

Correct option: D

none of these.

Option D (none of these) is correct because the calculated probability of getting a sum that is a perfect square is

\frac{17}{108}

, which is not among the given options.

12 Hard

12 of 23

Application +4 | -1

Three six-faced fair dice are thrown together. The probability that the sum of the numbers appearing on the dice is

k(3 \leq k \leq 8)

, is

View answer and explanation

Correct option: A

\frac{(k-1)(k-2)}{432}

Option A is correct because the probability is calculated by finding the coefficient of x^k in the generating function and dividing by the total number of outcomes.

13 Hard

13 of 23

Analyzing +4 | -1

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

View answer and explanation

Correct option: A

\frac{1}{6}

Option A (

\frac{1}{6}

) is the correct answer because it represents the calculated probability of at least 3 players picking the correct T-shirt using derangements.

14 Hard

14 of 23

Application +4 | -1

In a bag there are three tickets numbered 1, 2, 3. A ticket is drawn at random and put back, and this is done four times. The probability that the sum of the numbers is even, is

View answer and explanation

Correct option: C

40/81

Option C (40/81) is correct because it represents the probability that the sum of four tickets drawn with replacement is even.

15 Hard

15 of 23

Analyzing +4 | -1

In a convex hexagon two diagonals are drawn at random. The probability that the diagonals intersect at an interior point of the hexagon, is

View answer and explanation

Correct option: A

\frac{5}{12}

Option A (

\frac{5}{12}

) is correct because it represents the ratio of favorable outcomes (intersecting diagonals) to the total possible outcomes when selecting two diagonals.

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16 Hard

16 of 23

Application +4 | -1

An experiment has 10 equally likely outcomes. Let

A

and

B

be two non-empty events of the experiment. If

A

consists of 4 outcomes, the number of outcomes that

B

must have so that

A

and

B

are independent, is

View answer and explanation

Correct option: D

5 or 10

Option D (5 or 10) is correct because it identifies the number of outcomes for B that satisfy the independence condition P(A ∩ B) = P(A)P(B).

17 Hard

17 of 23

Analyzing +4 | -1

A ship is fitted with three engines

E_1, E_2

and

E_3

. The engines function independently of each other with respective probabilities

\frac{1}{2}, \frac{1}{4}

and

\frac{1}{4}

. For the ship to be operational at least two of its engines must function. Let

X

denote the event that ship is operational and let

X_1, X_2

and

X_3

denote respectively the events that the engines

E_1, E_2

and

E_3

are functioning. Which of the following is not true?

View answer and explanation

Correct option: C

P\left(X / X_2\right)=\frac{5}{16}

Option C (

P(X / X_2)=\frac{{5}}{{16}}

) is not true because the correct conditional probability

P(X/X_2)

\frac{{5}}{{8}}

, not

\frac{{5}}{{16}}

18 Hard

18 of 23

Application +4 | -1

Nine identical balls are numbers 1, 2, ...9. are put in a bag. A draws a ball and gets the number

a

. The ball is put back the bag. Next

B

draws a ball gets the number

b

. The probability that

a

and

b

satisfics the inequality

a-2 b+10 > 0

View answer and explanation

Correct option: C

\frac{61}{81}

Option C (

\frac{{61}}{{81}}

) is correct because it is the probability calculated by finding the number of pairs (a, b) that satisfy the inequality

a - 2b + 10 > 0

, divided by the total number of possible pairs.

19 Hard

19 of 23

Application +4 | -1

A dice is weighted such that the probability of rolling

\mathrm{n}

is proportional to

\mathrm{n}^2(\mathrm{n}=1,2,3,4,5,6)

. The dice is rolled twice, yielding the numbers a and

\mathrm{b}

. The probability that

\mathrm{a}<\mathrm{b}

in lowest form is

\frac{\mathrm{p}}{\mathrm{q}}

where

\frac{620}{(\mathrm{p}+\mathrm{q})}=

View answer and explanation

Correct option: A

Option A (5) is correct because the probability a < b is calculated by finding the probability of a=b and using symmetry to find P(a < b).

20 Hard

20 of 23

Application +4 | -1

Three dice are thrown. The probability that the sum of the number appearing is 15, is

View answer and explanation

Correct option: C

5/108

Option C (5/108) is correct because it represents the probability of the sum of three dice being 15, calculated using combinatorial methods.

21 Hard

21 of 23

Analyzing +4 | -1

A number

n

is randomly selected from the set

\{1,2,3, \ldots ., 1000\}

. The probability that

\frac{\sum_{i=1}^{n} i^{2}}{\sum_{i=1}^{n} i}

is an integer is

View answer and explanation

Correct option: C

0.334

Option C (0.334) is correct because it represents the probability that the given expression is an integer when n is randomly selected from the set.

22 Hard

22 of 23

Application +4 | -1

A four digit number (numbered from 0000 to 9999) is said to be lucky if the sum of its first two digits is equal to the sum of its last two digits. If a four digit number is picked up at random, then the probability that it is lucky is

View answer and explanation

Correct option: D

0.067

Option D (0.067) is correct because it represents the probability of a four-digit number being lucky, calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

23 Hard

23 of 23

Application +4 | -1

A game is played with fair cubic die which has one red side, two blue sides and three green sides. The result is the color of the top side after the dice is rolled. If the die is rolled repeatedly then the probability that second blue result occurs on or before the tenth roll can be expressed in the form of

\frac{3^p-2^q}{3^r}

where

p, q, r

are positive integers then the value of

p+r-q

View answer and explanation

Correct option: C

Option C (7) is correct because by calculating the probability of the second blue result occurring on or before the tenth roll and expressing it in the given form, we can determine the value of p+r-q.

Revision notes and key takeaways

Review PROBABILITY in your textbook or class notes, then use this set to check your understanding.

Key Takeaways

✓ This set contains 23 single-correct-answer MCQs on PROBABILITY.
✓ Try each question before opening its answer. Read the explanation where one is available.
✓ Sign in to save submitted answers. Revisit incorrect answers alongside your textbook or class notes.

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