PROBABILITY

Q. If A and B are two events such that P A ∩ B = 0.1 and P A B and

P B A are the roots of the equation 12𝑥 2 − 7𝑥 + 1 = 0, then the value
ഥ∪B
P A ഥ
of is:
ഥ∩B
P A ഥ
5 4 9 7
1 2 3 4
3 3 4 4
Sol. [JEE MAIN 2025]

Ans. (3)
Q. Two balls are selected at random one by one without replacement
from a bag containing 4 white and 6 black balls. If the probability that
the first selected ball is black, given that the second selected ball is also
𝑚
black, is , where gcd(𝑚, 𝑛) = 1, then 𝑚 + 𝑛 is equal to:
𝑛
1 14 2 4 3 11 (4) 13
Sol. [JEE MAIN 2025]

Ans. (1)
Q. A board has 16 squares as shown in the figure:

Out of these 16 squares, two squares are chosen at random.

The probability that they have no side in common is:
4 7 3 23
1 2 3 4
5 10 5 30
Sol. [JEE MAIN 2025]

Ans. (1)
Q. One die has two faces marked 1, two faces marked 2, one face marked 3
and one face marked 4. Another die has one face marked 1, two faces
marked 2, two faces marked 3 and one face marked 4. The probability
of getting the sum of numbers to be 4 or 5, when both the dice are
thrown together, is
1 3 2 4
1 2 3 4
2 5 3 9
Sol. [JEE MAIN 2025]

Ans. (1)
Q. Let A = [𝑎𝑖𝑗 ] be a square matrix of order 2 with entries either 0 or 1.
Let E be the event that A is an invertible matrix. Then the probability
P(E) is:
5 3 1 3
1 2 3 4
8 16 8 8
Sol. [JEE MAIN 2025]

Ans. (4)
Q. A and B alternately throw a pair of dice. A wins if he throws a sum of 5
before B throws a sum of 8, and B wins if he throws a sum of 8 before A
throws a sum of 5. The probability, that A wins if A makes the first
throw, is
9 9 8 8
1 2 3 4
17 19 17 19
Sol. [JEE MAIN 2025]

Ans. (2)
Q. Three defective oranges are accidently mixed with seven good ones and
on looking at them, it is not possible to differentiate between them. Two
oranges are drawn at random from the lot. If 𝑥 denote the number of
defective oranges, then the variance of 𝑥 is:
28 14 26 18
1 2 3 4
75 25 75 25
Sol. [JEE MAIN 2025]

Ans. (1)
Q. The coefficients 𝑎, 𝑏, 𝑐 in the quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 are
from the set {1, 2, 3, 4, 5, 6}. If the probability of this equation having
one real root bigger than the other is 𝑝, then 216𝑝 equals:
1 57 2 38 3 19 4 76
Sol. [JEE MAIN 2024]

Ans. (2)
Q. An integer is chosen at random from the integers 1, 2, 3, …, 50. The
probability that the chosen integer is a multiple of atleast one of 4, 6
and 7 is
8 21 9 14
1 2 3 4
25 50 50 25
Sol. [JEE MAIN 2024]

Ans. (2)
Q. Two integers 𝑥 and 𝑦 are chosen with replacement from the set
{0, 1, 2, 3, ….., 10}. Then the probability that |𝑥 − 𝑦| > 5 is:
30 62 60 31
1 2 3 4
121 121 121 121
Sol. [JEE MAIN 2024]

Ans. (1)
Q. Out of 11 consecutive natural numbers if three numbers are selected at
random (without repetition), then the probability that they are in A.P.
with positive common difference, is:
15 5 5 10
1 2 3 4
101 101 33 99
Sol. [JEE MAIN 2020]

Ans. (3)
Q. The probability of selecting integers 𝑎 ∈ −5, 30 such that
𝑥 2 + 2 𝑎 + 4 𝑥 − 5𝑎 + 64 > 0, for all 𝑥 ∈ ℝ, is:
7 2 1 1
1 2 3 4
36 9 6 4
Sol. [JEE MAIN 2021]

Ans. (2)
Q. A biased die is marked with numbers 2, 4, 8, 16, 32, 32 on its faces and
1
the probability of getting a face with mark 𝑛 is . If the die is thrown
𝑛
thrice, then the probability, that the sum of the numbers obtained is 48, is
7 7 3 13
1 11 2 12 3 10 4 12
2 2 2 2
Sol. [JEE MAIN 2022]

Ans. (4)
Q. Let N be the sum of the numbers appeared when two fair dice are
rolled and let the probability that N − 2, 3N, N + 2 are in geometric
𝑘
progression be . Then the value of 𝑘 is
48
1 2 2 4 3 16 4 8
Sol. [JEE MAIN 2023]

Ans. (2)
 2 1
Q. Let A and B be two events such that P B|A = , P A|B = and
5 7
1 5 1
P A ∩ B = . Consider S1 : P A′ ∪ B = , S2 : P A′ ∩ B ′ = .
9 6 18
Then
1 Both S1 and S2 are true 2 Both (S1) and (S2) are false
Sol. 3 Only S1 is true 4 Only (S2) is true
[JEE MAIN 2022]

Ans. (1)
Q. In a box, there are 20 cards, out of which 10 are labelled as A and the
remaining 10 are labelled as B. Cards are drawn at random, one after
the other and with replacement, till a second A-card is obtained. The
probability that the second A-card appears before the third B-card is:
11 13 9 15
1 2 3 4
16 16 16 16
Sol. [JEE MAIN 2020]

Ans. (1)
Q. Let A and B be independent events such that P(A) = 𝑝, P(B) = 2𝑝.
5
The largest value of 𝑝, for which P (exactly one of A, B occurs) = , is:
9
1 2 4 5
1 2 3 4
3 9 9 12
Sol. [JEE MAIN 2021]

Ans. (4)
Q. 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 = 𝜆𝑥 with one vertex
at the vertex of the parabola is________.
Sol. [JEE MAIN 2023]

Ans. (432)
Q. Let X be a binomially distributed random variable with mean 4 and
4
variance . Then 54 P(X ≤ 2) is equal to
3
73 146 146 126
1 2 3 4
27 27 81 81
Sol. [JEE MAIN 2022]

Ans. (2)