Probability

18 JEE Main questions · Maths

MathsJEE Main 21 Jan 2026, Shift 2·Q10·MCQ·Moderate
Open
Mean and variance of distribution
A random variable $X$ takes values 0, 1, 2, 3 with probabilities $\dfrac{2a+1}{30}, \dfrac{8a-1}{30}, \dfrac{4a+1}{30}$ , $b$ respectively, where $a, b \in \mathbb{R}$ . Let $\mu$ and $\sigma$ respectively be the mean and standard deviation of $X$ such that $\sigma^2 + \mu^2 = 2$ . Then $\dfrac{a}{b}$ is equal to:
1. (1)30
2. (2)3
3. (3)60
4. (4)12
MathsJEE Main 22 Jan 2026, Shift 1·Q4·MCQ·Easy–moderate
Open
Classical probability with divisibility
Two distinct numbers $a$ and $b$ are selected at random from $1, 2, 3, \ldots, 50$ . The probability, that their product $ab$ is divisible by 3, is
1. (1) $\dfrac{561}{1225}$
2. (2) $\dfrac{664}{1225}$
3. (3) $\dfrac{272}{1225}$
4. (4) $\dfrac{8}{25}$
MathsJEE Main 22 Jan 2026, Shift 1·Q7·MCQ·Easy–moderate
Open
Probability distribution of random variable
If a random variable $x$ has the probability distribution
$x$ 0 1 2 3 4 5 6 7
$p(x)$ 0 $2k$ $k$ $3k$ $2k^2$ $2k$ $k^2+k$ $7k^2$
then $P(3 < x \le 6)$ is equal to
1. (1)0.34
2. (2)0.22
3. (3)0.64
4. (4)0.33
MathsJEE Main 23 Jan 2026, Shift 1·Q24·Numerical·Moderate
Open
Classical probability counting
From the first 100 natural numbers, two numbers first $a$ and then $b$ are selected randomly without replacement. If the probability that $a - b \ge 10$ is $\dfrac{m}{n}$ , $\gcd(m, n) = 1$ , then $m + n$ is equal to .
MathsJEE Main 23 Jan 2026, Shift 2·Q11·MCQ·Easy–moderate
Open
Total probability theorem
Bag A contains 9 white and 8 black balls, while bag B contains 6 white and 4 black balls. One ball is randomly picked up from the bag B and mixed up with the balls in the bag A. Then a ball is randomly drawn from the bag A. If the probability, that the ball drawn is white, is $\dfrac{p}{q}$ , $\gcd(p, q) = 1$ , then $p + q$ is equal to
1. (1)22
2. (2)23
3. (3)24
4. (4)21
MathsJEE Main 24 Jan 2026, Shift 1·Q17·MCQ·Easy–moderate
Open
Binomial distribution
From a lot containing 10 defective and 90 non-defective bulbs, 8 bulbs are selected one by one with replacement. Then the probability of getting at least 7 defective bulbs is :-
1. (1) $\dfrac{7}{10^7}$
2. (2) $\dfrac{81}{10^8}$
3. (3) $\dfrac{67}{10^8}$
4. (4) $\dfrac{73}{10^8}$
MathsJEE Main 24 Jan 2026, Shift 2·Q24·Numerical·Moderate
Open
Classical probability with subsets
Let $S$ be a set of 5 elements and $P(S)$ denote the power set of $S$ . Let $E$ be an event of choosing an ordered pair $(A, B)$ from the set $P(S)\times P(S)$ such that $A \cap B = \varnothing$ . If the probability of the event $E$ is $\dfrac{3^p}{2^q}$ , where $p, q \in \mathbb{N}$ , then $p + q$ is equal to .
MathsJEE Main 28 Jan 2026, Shift 1·Q9·MCQ·Moderate
Open
Bayes theorem
A bag contains 10 balls out of which $k$ are red and $(10 - k)$ are black, where $0 \le k \le 10$ . If three balls are drawn at random without replacement and all of them are found to be black, then the probability that the bag contains 1 red and 9 black balls is:
1. (1) $\dfrac{7}{11}$
2. (2) $\dfrac{7}{55}$
3. (3) $\dfrac{7}{110}$
4. (4) $\dfrac{14}{55}$
We corrected a printing error in the original paper
Source printed “the bag contains l red and 9 black balls” (letter l for the digit 1).

MathsJEE Main 28 Jan 2026, Shift 2·Q4·MCQ·Easy–moderate

Expectation of a random variable

The probability distribution of a random variable

X

is given below :

$X$	$4k$	$\dfrac{30}{7}k$	$\dfrac{32}{7}k$	$\dfrac{34}{7}k$	$\dfrac{36}{7}k$	$\dfrac{38}{7}k$	$\dfrac{40}{7}k$	$6k$
$P(X)$	$\dfrac{2}{15}$	$\dfrac{1}{15}$	$\dfrac{2}{15}$	$\dfrac{1}{5}$	$\dfrac{1}{15}$	$\dfrac{2}{15}$	$\dfrac{1}{5}$	$\dfrac{1}{15}$

E(X) = \dfrac{263}{15}

, then

P(X < 20)

is equal to :

(1) $\dfrac35$
(2) $\dfrac{8}{15}$
(3) $\dfrac{11}{15}$
(4) $\dfrac{14}{15}$

MathsJEE Main 2 Apr 2026, Shift 1·Q23·Numerical·Moderate
Open
Classical probability counting
Let $a, b, c \in \{1, 2, 3, 4\}$ . If the probability, that $ax^2 + 2\sqrt2\,bx + c > 0$ for all $x \in \mathbb{R}$ , is $\dfrac{m}{n}$ , $\gcd(m, n) = 1$ , then $m + n$ is equal to .
MathsJEE Main 2 Apr 2026, Shift 2·Q8·MCQ·Moderate
Open
Probability of reaching a score
A man throws a fair coin repeatedly. He gets 10 points for each head he throws and 5 points for each tail he throws. If the probability that he gets exactly 30 points is $\dfrac{m}{n}$ , $\gcd(m, n) = 1$ , then $m + n$ is equal to:
1. (1)53
2. (2)55
3. (3)107
4. (4)105
MathsJEE Main 4 Apr 2026, Shift 1·Q21·Numerical·Moderate
Open
Independent trials counting outcomes
A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six tosses and exactly 3 heads appear in the last five tosses is $p$ , then $96p$ is equal to .
MathsJEE Main 4 Apr 2026, Shift 2·Q21·Numerical·Moderate
Open
Classical probability
From a month of 31 days, 3 different dates are selected at random. If the probability that these dates are in an increasing A.P. is equal to $\dfrac{a}{b}$ , where $a, b \in \mathbb{N}$ and $\gcd(a, b) = 1$ , then $a + b$ is equal to .
MathsJEE Main 5 Apr 2026, Shift 1·Q7·MCQ·Moderate
Open
Bayes theorem
A letter is known to have arrived by post either from KANPUR or from ANANTPUR. On the envelope just two consecutive letters AN are visible. The probability, that the letter came from ANANTPUR, is:
1. (1) $\dfrac{7}{10}$
2. (2) $\dfrac{10}{17}$
3. (3) $\dfrac{12}{19}$
4. (4) $\dfrac{7}{19}$
We corrected a printing error in the original paper
Source printed “On the the envelope” (doubled word).
MathsJEE Main 5 Apr 2026, Shift 2·Q8·MCQ·Easy–moderate
Open
Total probability theorem
The probabilities that players $A$ and $B$ of a team are selected for the captaincy for a tournament are 0.6 and 0.4, respectively. If $A$ is selected the captain, the probability that the team wins the tournament is 0.8 and if $B$ is selected the captain, the probability that the team wins the tournament is 0.7. Then the probability, that the team wins the tournament, is:
1. (1)0.74
2. (2)0.76
3. (3)0.72
4. (4)0.78
MathsJEE Main 6 Apr 2026, Shift 1·Q10·MCQ·Easy–moderate
Open
Total probability theorem
A bag contains $(N + 1)$ coins – $N$ fair coins, and one coin with ‘Head' on both sides. A coin is selected at random and tossed. If the probability of getting ‘Head' is $\dfrac{9}{16}$ , then $N$ is equal to:
1. (1)5
2. (2)7
3. (3)8
4. (4)9
MathsJEE Main 6 Apr 2026, Shift 2·Q9·MCQ·Moderate
Open
Classical probability of pairings
A bag contains 6 blue and 6 green balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each drawn pair consists of one blue and one green ball is:
1. (1) $\dfrac{63}{925}$
2. (2) $\dfrac{17}{231}$
3. (3) $\dfrac{16}{231}$
4. (4) $\dfrac{64}{925}$
MathsJEE Main 8 Apr 2026, Shift 2·Q6·MCQ·Easy–moderate
Open
Bayes theorem
A candidate has to go to the examination centre to appear in an examination. The candidate uses only one means of transportation for the entire distance out of bus, scooter and car. The probabilities of the candidate going by bus, scooter and car, respectively, are $\dfrac25$ , $\dfrac15$ and $\dfrac25$ . The probabilities that the candidate reaches late at the examination centre are $\dfrac15$ , $\dfrac13$ and $\dfrac14$ if the candidate uses bus, scooter and car, respectively. Given that the candidate reached late at the examination centre, the probability that the candidate travelled by bus is:
1. (1) $\dfrac{11}{37}$
2. (2) $\dfrac{12}{37}$
3. (3) $\dfrac{13}{37}$
4. (4) $\dfrac{14}{37}$

$x$	0	1	2	3	4	5	6	7
$p(x)$	0	$2k$	$k$	$3k$	$2k^2$	$2k$	$k^2+k$	$7k^2$