Probability — JEE Main PYQs

In a bombing attack, there is $50\%$ chance that a bomb will hit the target. At least two independent hits are required to destroy the target completely. Then the minimum number of bombs, that must be dropped to ensure that there is at least $99\%$ chance of completely dest …

A die is thrown two times and the sum of the scorers 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

The probability that a randomly chosen 5 - digit number is made from exactly two digits is:

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 …

Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five, is ______.

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:

The probability of a man hitting a target is $\frac{1}{10}$. The least number of shots required, so that the probability of his hitting the target at least once is greater than $\frac{1}{4}$, is ______ .

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 a total of 6 before $B$ throws a total of 7 and $B$ wins the game if he throws a total of …

Let A and b be two non - null events such that $\mathrm{A}\subset \mathrm{B}$. Then, which of the following statements is always correct?

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:

Four persons can hit a target correctly with probabilities $\frac{1}{2},\frac{1}{3},\frac{1}{4}$ and $\frac{1}{8}$ respectively. If all hit at the target independently, then the probability that the target would be hit, is:

An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on the is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered $1,2,3,\dots \dots ,9$ is rando …

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 $\left(\frac{\text{mean of }X}{\text{standard deviation of }X}\right)$ is equal to:

Let $S=\{1,2,\dots ,20\}$. 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 randomly chosen subset of $S$ is 'nice' is:

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:

Let a random variable $X$ have a binomial distribution with mean 8 and variance 4. If $\mathrm{P}(\mathrm{X}\le 2)=\frac{\mathrm{k}}{2^{16}}$, then k is equal to:

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

Two integers are selected at random from the set $\{1,2,\dots ,11\}$. Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is:

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:

For an initial screening of an admission test, a candidate is given fifty problems to solve. If the probability that the candidate can solve any problem is $\frac{4}{5}$, then the probability that he is unable to solve less than two problems is :

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 :

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 :

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 student selected has opted neither for NCC nor for NSS is :

Two newspaper A and B are published in a city. It is known that $25\%$ of the city populations reads A and $20\%$ reads B while $8\%$ reads both A and B. Further, $30\%$ of those who read A but not B look into advertisements and $40\%$ of those who read B but not A also look into …

An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn; the original ball is not returned to the urn. Now, a second ball i …

Two cards are drawn successively with replacement from a well shuffled deck of 52 cards. Let $X$ denote the random variable of number of aces obtained in the two drawn cards. Then $\mathrm{P}(\mathrm{X}=1)+\mathrm{P}(\mathrm{X}=2)$ equals:

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 bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with two additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn bal …

A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is :

If two different numbers are taken from the set $\{0,1,2,3,\dots \dots ,10\}$; then the probability that their sum as well as absolute difference are both multiple of 4, is :

For three events $\mathrm{A},\mathrm{B}$ and C ,

P(Exactly one of A or B occurs)
$=\mathrm{P}($ Exactly one of B or C occurs $)$
$=\mathrm{P}($ Exactly one of C or A occurs $)=\frac{1}{4}$ and
P(All the three events occur simultaneously) $=\frac{1}{16}$.

Then the p …

Let two fair six-faced dice A and B be thrown simultaneously. If $E_1$ is the event that die A shows up four, $E_2$ is the event that die B shows up two and $E_3$ is the event that the sum of numbers on both dice is odd, then which of the following statements is NOT t …

Let A and B be two events such that $P(\overline{A\cup B})=\frac{1}{6},P(A\cap B)=\frac{1}{4}$ and $P(\bar{A})=\frac{1}{4}$, where $\bar{A}$ stands for the complement of the event A. Then the events A and B are

A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get 4 or more correct answers just by guessing is

Three numbers are chosen at random without replacement from $\{1,2,3,\dots \dots 8\}$. The probability that their minimum is 3, given that their maximum is 6, is

Consider 5 independent Bernoulli's trials each with probability of success p. If the probability of at least one failure is greater than or equal to $\frac{31}{32}$, then $p$ lies in the interval

If C and D are two events such that $\mathrm{C}\subset \mathrm{D}$ and $\mathrm{P}(\mathrm{D})\ne 0$, then the correct statement among the following is

Four numbers are chosen at random (without replacement) from the set $\{1,2,3,\dots ,20\}$.

Statement-1: The probability that the chosen numbers when arranged in some order will form an AP is $\frac{1}{85}$.
Statement-2: If the four chosen numbers from an AP, then the set …

An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colour is

In a binomial distribution $B\left(n,p=\frac{1}{4}\right)$, if the probability of at least one success is greater than or equal to $\frac{9}{10}$, then n is greater than

One ticket is selected at random from 50 tickets numbered $00,01,02,\dots .,49$. Then the probability that the sum of the digits on the selected ticket is 8, given that the product of these digits is zero, equals

It is given that the events $A$ and $B$ are such that $P(A)=\frac{1}{4},P\left(\frac{A}{B}\right)=\frac{1}{2}$ and $P\left(\frac{B}{A}\right)=\frac{2}{3}$. Then $P(B)$ is

A die is thrown. Let A be the event that the number obtained is greater than 3. Let B be the event that the number obtained is less than 5. Then $P(A\cup B)$ is

A pair of fair dice is thrown independently three times. The probability of getting a score of exactly 9 twice is

Two aeroplanes I and II bomb a target in succession. The probabilities of I and II scoring a hit correctly are 0.3 and 0.2, respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is

At a telephone enquiry system the number of phone cells regarding relevant enquiry follow Poisson distribution with an average of 5 phone calls during 10-minute time intervals. The probability that there is at the most one phone call during a 10-minute time period is

Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is

A random variable $X$ has Poisson distribution with mean 2. Then $P(X>1.5)$ equals

Let $A$ and $B$ be two events such that $P(\overline{A\cup B})=\frac{1}{6},P(A\cap B)=\frac{1}{4}$ and $P(\bar{A})=\frac{1}{4}$, where $\bar{A}$ stands for complement of event $A$. Then events $A$ and $B$ are

The probability that $A$ speaks truth is $\frac{4}{5}$, while this probability for $B$ is $\frac{3}{4}$. The probability that they contradict each other when asked to speak on a fact is