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Probability

365 Questions
All Exams JEE Mains JEE Advanced
2021 JEE Mains MCQ
JEE Main 2021 (Online) 27th July Evening Shift
A student appeared in an examination consisting of 8 true-false type questions. The student guesses the answers with equal probability. the smallest value of n, so that the probability of guessing at least 'n' correct answers is less than ${1 \over 2}$, is :
A.
5
B.
6
C.
3
D.
4
2021 JEE Mains MCQ
JEE Main 2021 (Online) 27th July Morning Shift
The probability that a randomly selected 2-digit number belongs to the set {n $\in$ N : (2n $-$ 2) is a multiple of 3} is equal to :
A.
${1 \over 6}$
B.
${2 \over 3}$
C.
${1 \over 2}$
D.
${1 \over 3}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 25th July Evening Shift
Let X be a random variable such that the probability function of a distribution is given by $P(X = 0) = {1 \over 2},P(X = j) = {1 \over {{3^j}}}(j = 1,2,3,...,\infty )$. Then the mean of the distribution and P(X is positive and even) respectively are :
A.
${3 \over 8}$ and ${1 \over 8}$
B.
${3 \over 4}$ and ${1 \over 8}$
C.
${3 \over 4}$ and ${1 \over 9}$
D.
${3 \over 4}$ and ${1 \over 16}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 25th July Morning Shift
Let 9 distinct balls be distributed among 4 boxes, B1, B2, B3 and B4. If the probability than B3 contains exactly 3 balls is $k{\left( {{3 \over 4}} \right)^9}$ then k lies in the set :
A.
{x $\in$ R : |x $-$ 3| < 1}
B.
{x $\in$ R : |x $-$ 2| $\le$ 1}
C.
{x $\in$ R : |x $-$ 1| < 1}
D.
{x $\in$ R : |x $-$ 5| $\le$ 1}
2021 JEE Mains MCQ
JEE Main 2021 (Online) 22th July Evening Shift
Four dice are thrown simultaneously and the numbers shown on these dice are recorded in 2 $\times$ 2 matrices. The probability that such formed matrix have all different entries and are non-singular, is :
A.
${{45} \over {162}}$
B.
${{21} \over {81}}$
C.
${{22} \over {81}}$
D.
${{43} \over {162}}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 20th July Evening Shift
Let A, B and C be three events such that the probability that exactly one of A and B occurs is (1 $-$ k), the probability that exactly one of B and C occurs is (1 $-$ 2k), the probability that exactly one of C and A occurs is (1 $-$ k) and the probability of all A, B and C occur simultaneously is k2, where 0 < k < 1. Then the probability that at least one of A, B and C occur is :
A.
greater than ${1 \over 8}$ but less than ${1 \over 4}$
B.
greater than ${1 \over 2}$
C.
greater than ${1 \over 4}$ but less than ${1 \over 2}$
D.
exactly equal to ${1 \over 2}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 20th July Morning Shift
Words with or without meaning are to be formed using all the letters of the word EXAMINATION. The probability that the letter M appears at the fourth position in any such word is :
A.
${1 \over {66}}$
B.
${1 \over {11}}$
C.
${1 \over {9}}$
D.
${2 \over {11}}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 20th July Morning Shift
The probability of selecting integers a$\in$[$-$ 5, 30] such that x2 + 2(a + 4)x $-$ 5a + 64 > 0, for all x$\in$R, is :
A.
${7 \over {36}}$
B.
${2 \over {9}}$
C.
${1 \over {6}}$
D.
${1 \over {4}}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 18th March Evening Shift
Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be 0.4096 and 0.2048 respectively. Then the probability of getting exactly 3 successes is equal to :
A.
${{40} \over {243}}$
B.
${{128} \over {625}}$
C.
${{80} \over {243}}$
D.
${{32} \over {625}}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 17th March Evening Shift
Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with probability of occurrence of 0 at even places be ${1 \over 2}$ and probability of occurrence of 0 at the odd place be ${1 \over 3}$. Then the probability that '10' is followed by '01' is equal to :
A.
${1 \over 18}$
B.
${1 \over 3}$
C.
${1 \over 9}$
D.
${1 \over 6}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 17th March Morning Shift
Two dies are rolled. If both dices have six faces numbered 1, 2, 3, 5, 7 and 11, then the probability that the sum of the numbers on the top faces is less than or equal to 8 is :
A.
${4 \over 9}$
B.
${1 \over 2}$
C.
${5 \over {12}}$
D.
${17 \over {36}}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 16th March Evening Shift
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.
${4 \over {9}}$
B.
${9 \over {56}}$
C.
${11 \over {27}}$
D.
${3 \over {7}}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 16th March Morning Shift
A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is :
A.
${{39} \over {50}}$
B.
${{3} \over {4}}$
C.
${{22} \over {425}}$
D.
${{52} \over {867}}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th February Evening Shift
A seven digit number is formed using digits 3, 3, 4, 4, 4, 5, 5. The probability, that number so formed is divisible by 2, is :
A.
${1 \over 7}$
B.
${4 \over 7}$
C.
${6 \over 7}$
D.
${3 \over 7}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th February Morning Shift
A fair coin is tossed a fixed number of times. If the probability of getting 7 heads is equal to probability of getting 9 heads, then the probability of getting 2 heads is :
A.
${{15} \over {{2^8}}}$
B.
${{15} \over {{2^{12}}}}$
C.
${{15} \over {{2^{13}}}}$
D.
${{15} \over {{2^{14}}}}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 25th February Evening Shift
Let A be a set of all 4-digit natural numbers whose exactly one digit is 7. Then the probability that a randomly chosen element of A leaves remainder 2 when divided by 5 is :
A.
${2 \over 9}$
B.
${1 \over 5}$
C.
${122 \over 297}$
D.
${97 \over 297}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 25th February Evening Shift
In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the remaining 140 are non-smokers and vegetarian. Their chances of getting a particular chest disorder are 35%, 20% and 10% respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is :
A.
${{14} \over {45}}$
B.
${{8} \over {45}}$
C.
${{7} \over {45}}$
D.
${{28} \over {45}}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 25th February Morning Shift
When a missile is fired from a ship, the probability that it is intercepted is ${1 \over 3}$ and the probability that the missile hits the target, given that it is not intercepted, is ${3 \over 4}$. If three missiles are fired independently from the ship, then the probability that all three hit the target, is :
A.
${3 \over 4}$
B.
${3 \over 8}$
C.
${1 \over 27}$
D.
${1 \over 8}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 25th February Morning Shift
The coefficients a, b and c of the quadratic equation, ax2 + bx + c = 0 are obtained by throwing a dice three times. The probability that this equation has equal roots is :
A.
${1 \over {72}}$
B.
${5 \over {216}}$
C.
${1 \over {36}}$
D.
${1 \over {54}}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 24th February Evening Shift
The probability that two randomly selected subsets of the set {1, 2, 3, 4, 5} have exactly two elements in their intersection, is :
A.
${{135} \over {{2^9}}}$
B.
${{65} \over {{2^8}}}$
C.
${{65} \over {{2^7}}}$
D.
${{35} \over {{2^7}}}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 24th February Morning Shift
An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times, then the probability of getting an odd number for odd number of times is :
A.
${5 \over {36}}$
B.
${3 \over {16}}$
C.
${1 \over 2}$
D.
${1 \over {32}}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 6th September Evening Slot
The probabilities of three events A, B and C are given by
P(A) = 0.6, P(B) = 0.4 and P(C) = 0.5.
If P(A$ \cup $B) = 0.8, P(A$ \cap $C) = 0.3, P(A$ \cap $B$ \cap $C) = 0.2, P(B$ \cap $C) = $\beta $
and P(A$ \cup $B$ \cup $C) = $\alpha $, where 0.85 $ \le \alpha \le $ 0.95, then $\beta $ lies in the interval :
A.
[0.35, 0.36]
B.
[0.20, 0.25]
C.
[0.25, 0.35]
D.
[0.36, 0.40]
2020 JEE Mains MCQ
JEE Main 2020 (Online) 6th September Morning Slot
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 :
A.
${{10} \over {99}}$
B.
${{5} \over {33}}$
C.
${{15} \over {101}}$
D.
${{5} \over {101}}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 4th September Evening Slot
In a game two players A and B take turns in throwing a pair of fair dice starting with player A and total of scores on the two dice, in each throw is noted. A wins the game if he throws total a of 6 before B throws a total of 7 and B wins the game if he throws a total of 7 before A throws a total of six. The game stops as soon as either of the players wins. The probability of A winning the game is :
A.
${5 \over {6}}$
B.
${5 \over {31}}$
C.
${31 \over {61}}$
D.
${30 \over {61}}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 3rd September Evening Slot
The probability that a randomly chosen 5-digit number is made from exactly two digits is :
A.
${{150} \over {{{10}^4}}}$
B.
${{134} \over {{{10}^4}}}$
C.
${{121} \over {{{10}^4}}}$
D.
${{135} \over {{{10}^4}}}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 3rd September Morning Slot
A dice is thrown two times and the sum of the scores appearing on the die is observed to be a multiple of 4. Then the conditional probability that the score 4 has appeared atleast once is :
A.
${1 \over 8}$
B.
${1 \over 9}$
C.
${1 \over 4}$
D.
${1 \over 3}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 2nd September Evening Slot
Let EC denote the complement of an event E. Let E1 , E2 and E3 be any pairwise independent events with P(E1) > 0

and P(E1 $ \cap $ E2 $ \cap $ E3) = 0.

Then P($E_2^C \cap E_3^C/{E_1}$) is equal to :
A.
$P\left( {E_3^C} \right)$ - P(E2)
B.
$P\left( {E_2^C} \right)$ + P(E3)
C.
$P\left( {E_3^C} \right)$ - $P\left( {E_2^C} \right)$
D.
P(E3) - $P\left( {E_2^C} \right)$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 2nd September Morning Slot
Box I contains 30 cards numbered 1 to 30 and Box II contains 20 cards numbered 31 to 50. A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box I is :
A.
${8 \over {17}}$
B.
${2 \over 3}$
C.
${2 \over 5}$
D.
${4 \over {17}}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 9th January Evening Slot
If 10 different balls are to be placed in 4 distinct boxes at random, then the probability that two of these boxes contain exactly 2 and 3 balls is :
A.
${{965} \over {{2^{11}}}}$
B.
${{965} \over {{2^{10}}}}$
C.
${{945} \over {{2^{11}}}}$
D.
${{945} \over {{2^{10}}}}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 9th January Evening Slot
A random variable X has the following probability distribution :

X: 1 2 3 4 5
P(X): K2 2K K 2K 5K2

Then P(X > 2) is equal to :
A.
${1 \over {6}}$
B.
${7 \over {12}}$
C.
${1 \over {36}}$
D.
${23 \over {36}}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 9th January Morning Slot
In a box, there are 20 cards, out of which 10 are lebelled 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 :
A.
${{13} \over {16}}$
B.
${{11} \over {16}}$
C.
${{15} \over {16}}$
D.
${{9} \over {16}}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 8th January Evening Slot
Let A and B be two events such that the probability that exactly one of them occurs is ${2 \over 5}$ and the probability that A or B occurs is ${1 \over 2}$ , then the probability of both of them occur together is :
A.
0.20
B.
0.02
C.
0.01
D.
0.10
2020 JEE Mains MCQ
JEE Main 2020 (Online) 8th January Morning Slot
Let A and B be two independent events such that
P(A) = ${1 \over 3}$ and P(B) = ${1 \over 6}$.
Then, which of the following is TRUE?
A.
$P\left( {{A \over {A \cup B}}} \right) = {1 \over 4}$
B.
$P\left( {{A \over B}} \right) = {2 \over 3}$
C.
$P\left( {{{A'} \over {B'}}} \right) = {1 \over 3}$
D.
$P\left( {{A \over {B'}}} \right) = {1 \over 3}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 7th January Evening Slot
In a workshop, there are five machines and the probability of any one of them to be out of service on a day is ${{1 \over 4}}$ . If the probability that at most two machines will be out of service on the same day is ${\left( {{3 \over 4}} \right)^3}k$, then k is equal to :
A.
${{{17} \over 4}}$
B.
${{{17} \over 2}}$
C.
${{{17} \over 8}}$
D.
4
2020 JEE Mains MCQ
JEE Main 2020 (Online) 7th January Morning Slot
An unbiased coin is tossed 5 times. Suppose that a variable X is assigned the value of k when k consecutive heads are obtained for k = 3, 4, 5, otherwise X takes the value -1. Then the expected value of X, is :
A.
$ - {3 \over {16}}$
B.
$ - {1 \over 8}$
C.
${1 \over 8}$
D.
${3 \over {16}}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th April Evening Slot
A person throws two fair dice. He wins Rs. 15 for throwing a doublet (same numbers on the two dice), wins Rs. 12 when the throw results in the sum of 9, and loses Rs. 6 for any other outcome on the throw. Then the expected gain/loss (in Rs.) of the person is :
A.
${1 \over 4}$ loss
B.
${1 \over 2}$ gain
C.
${1 \over 2}$ loss
D.
2 gain
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th April Evening Slot
For an initial screening of an admission test, a candidate is given fifty problems to solve. If the probability that the candidate solve any problem is ${4 \over 5}$ , then the probability that he is unable to solve less than two problems is :
A.
${{164} \over {25}}{\left( {{1 \over 5}} \right)^{48}}$
B.
${{316} \over {25}}{\left( {{4 \over 5}} \right)^{48}}$
C.
${{201} \over 5}{\left( {{1 \over 5}} \right)^{49}}$
D.
${{54} \over 5}{\left( {{4 \over 5}} \right)^{49}}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th April Morning Slot
If three of the six vertices of a regular hexagon are chosen at random, then the probability that the triangle formed with these chosen vertices is equilateral is :
A.
${1 \over {10}}$
B.
${3 \over {10}}$
C.
${3 \over {20}}$
D.
${1 \over {5}}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th April Morning Slot
Let a random variable X have a binomial distribution with mean 8 and variance 4. If $P\left( {X \le 2} \right) = {k \over {{2^{16}}}}$, then k is equal to :
A.
17
B.
1
C.
137
D.
121
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th April Evening Slot
Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is more than 99% is :
A.
6
B.
5
C.
8
D.
7
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th April Morning Slot
Assume that each born child is equally likely to be a boy or a girl. If two families have two children each, then the conditional probability that all children are girls given that at least two are girls is :
A.
${1 \over {10}}$
B.
${1 \over {17}}$
C.
${1 \over {11}}$
D.
${1 \over {12}}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 9th April Morning Slot
Four persons can hit a target correctly with probabilities ${1 \over 2}$, ${1 \over 3}$, ${1 \over 4}$ and ${1 \over 8}$ respectively. if all hit at the target independently, then the probability that the target would be hit, is :
A.
${{25} \over {32}}$
B.
${{25} \over {192}}$
C.
${{1} \over {192}}$
D.
${{7} \over {32}}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 8th April Evening Slot
The minimum number of times one has to toss a fair coin so that the probability of observing at least one head is at least 90% is :
A.
2
B.
3
C.
4
D.
5
2019 JEE Mains MCQ
JEE Main 2019 (Online) 8th April Morning Slot
Let A and B be two non-null events such that A $ \subset $ B . Then, which of the following statements is always correct?
A.
P(A|B) = 1
B.
P(A|B) = P(B) – P(A)
C.
P(A|B) $ \le $ P(A)
D.
P(A|B) $ \ge $ P(A)
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th January Evening Slot
In a class of 60 students, 40 opted for NCC, 30 opted for NSS and 20 opted for both NCC and NSS. If one of these students is selected at random, then the probability that the students selected has opted neither for NCC nor for NSS is :
A.
${1 \over 3}$
B.
${1 \over 6}$
C.
${2 \over 3}$
D.
${5 \over 6}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th January Evening Slot
In a game, a man wins Rs. 100 if he gets 5 or 6 on a throw of a fair die and loses Rs. 50 for getting any other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum of three throws, then his expected gain/loss (in rupees) is :
A.
${{400} \over 3}$ loss
B.
0
C.
${{400} \over 9}$ loss
D.
${{400} \over 3}$ gain
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th January Morning Slot
In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to :
A.
${{200} \over {{6^5}}}$
B.
${{225} \over {{6^5}}}$
C.
${{150} \over {{6^5}}}$
D.
${{175} \over {{6^5}}}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 11th January Evening Slot
Let  S = {1, 2, . . . . . ., 20}. A subset B of S is said to be "nice", if the sum of the elements of B is 203. Then the probability that a randonly chosen subset of S is "nice" is :
A.
${5 \over {{2^{20}}}}$
B.
${7 \over {{2^{20}}}}$
C.
${4 \over {{2^{20}}}}$
D.
${6 \over {{2^{20}}}}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 11th January Evening Slot
A bag contains 30 white balls and 10 red balls. 16 balls are drawn one by one randomly from the bag with replacement. If X be the number of white balls drawn, then $\left( {{{mean\,\,of\,X} \over {s\tan dard\,\,deviation\,\,of\,X}}} \right)$ is equal to :
A.
4
B.
$3\sqrt 2 $
C.
${{4\sqrt 3 } \over 3}$
D.
$4\sqrt 3 $
2019 JEE Mains MCQ
JEE Main 2019 (Online) 11th January Morning Slot
Two integers are selected at random from the set {1, 2, ...., 11}. Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is :
A.
${2 \over 5}$
B.
${1 \over 2}$
C.
${7 \over 10}$
D.
${3 \over 5}$