Probability — JEE Main PYQs

The probability that a randomly chosen one-one function from the set $\{\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\}$ to the set $\{1,2,3,4,5\}$ satisfies $\mathrm{f}(\mathrm{a})+2\mathrm{f}(\mathrm{b})-\mathrm{f}(\mathrm{c})=\mathrm{f}(\mathrm{d})$ is :

If a point $A(x,y)$ lies in the region bounded by the y-axis, straight lines $2y+x=6$ and $5x-6y=30$, then the probability that $y<1$ is :

The probability, that in a randomly selected 3-digit number at least two digits are odd, is

If the probability that a randomly chosen 6-digit number formed by using digits 1 and 8 only is a multiple of 21 is $p$, then $96p$ is equal to ____.

The probability that a randomly chosen $2\times 2$ matrix with all the entries from the set of first 10 primes, is singular, is equal to :

Let a biased coin be tossed 5 times. If the probability of getting 4 heads is equal to the probability of getting 5 heads, then the probability of getting atmost two heads is:

Five numbers ${\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4,{\mathrm{x}}_5$ are randomly selected from the numbers $1,2,3,\dots ,18$ and are arranged in the increasing order ( ${\mathrm{x}}_1<{\mathrm{x}}_2<{\mathrm{x}}_3<{\mathrm{x}}_4<{\mathrm{x}}_5$ ) …

Let X be a random variable having binomial distribution $\mathrm{B}(7,\mathrm{p})$. If $\mathrm{P}(\mathrm{X}=3)=5\mathrm{P}(\mathrm{X}=4)$, then the sum of the mean and the variance of X is :

Let $\mathrm{S}=\{1,2,3,\dots ,2022\}$. Then the probability, that a randomly chosen number n from the set S such that $\mathrm{HCF}(\mathrm{n},2022)=1$, is :

Bag A contains 2 white, 1 black and 3 red balls and bag B contains 3 black, 2 red and n white balls. One bag is chosen at random and 2 balls drawn from it at random, are found to be 1 red and 1 black. If the probability that both balls come from Bag A is $\frac{6}{11}$, then n …

If a random variable X follows the Binomial distribution $\mathrm{B}(33,\mathrm{p})$ such that $3\mathrm{P}(\mathrm{X}=0)=\mathrm{P}(\mathrm{X}=1)$, then the value of …

A random variable $X$ has the following probability distribution:

The value of $P(1<X<4\mid X\le 2)$ is equal to:

In an examination, there are 10 true-false type questions. Out of 10, a student can guess the answer of 4 questions correctly with probability $\frac{3}{4}$ and the remaining 6 questions correctly with probability $\frac{1}{4}$.

If the probability that the student guesses the an …

A biased die is marked with numbers $2,4,8,16$, 32, 32 on its faces and the probability of getting a face with mark $n$ is $\frac{1}{n}$. If the die is thrown thrice, then the probability, that the sum of the numbers obtained is 48 , is

Let $E_1$ and $E_2$ be two events such that the conditional probabilities $\mathrm{P}\left({\mathrm{E}}_1\mid {\mathrm{E}}_2\right)=\frac{1}{2}$, $\mathrm{P}\left({\mathrm{E}}_2\mid {\mathrm{E}}_1\right)=\frac{3}{4}$ and …

Let $A$ and $B$ be independent events such that $P(A)=p,P(B)=2p$. The largest value of $p$, for which $P$ (exactly one of $A,B$ occurs) $=\frac{5}{9}$, is:

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 of times is:

Let $B_i(i=1,2,3)$ be three independent events in a sample space. The probability that only $B_1$ occur is $\alpha$, only $B_2$ occurs is $\beta$ and only $B_3$ occurs is $\gamma$. Let $p$ be the probability that none of the events $B_i$ occurs and these 4 probabilities satisfy t …

A fair die is tossed until six is obtained on it. Let $X$ be the number of required tosses, then the conditional probability $P(X\ge 5\mid X>2)$ is:

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 :

The probability that two randomly selected subsets of the set $\{1,2,3,4,5\}$ have exactly two elements in their intersection is:

Let $\mathrm{A},\mathrm{B}$ 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 $(1-\mathrm{k})$ and the probabil …

Each of the persons $A$ and $B$ independently tosses three fair coins. The probability that both of them get the same number of heads is:

The probability distribution of random variable $X$ is given by:

Let $p=P(1<X<4\mid X<3)$. If $5p=\lambda \cdot k$ then $\lambda$ is equal to

When a missile is fired from a ship, the probability that it is intercepted is $\frac{1}{3}$ and the probability that the missile hits the target, given that it is not intercepted, is $\frac{3}{4}$. If three missiles are fired independently from the ship, then probability that al …

The coefficients $a,b$ and $c$ of the quadratic equation, $a{x}^2+bx+c=0$ are obtained by throwing a dice three times. The probability that this equation has equal roots is :

Four dice are thrown simultaneously and the numbers shown on these dice are recorded in $2\times 2$ matrices. The probability that such formed matrices have all different entries and are non-singular, is:

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 :

When a certain biased die is rolled, a particular face occurs with probability $\frac{1}{6}-x$ and its opposite face occurs with probability $\frac{1}{6}+x$. All other faces occur with probability $\frac{1}{6}$. Note that opposite faces sum to 7 in any die. If $0<x<\frac{1}{6}$, …

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 :

An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is 0.9 and that of the second unit is 0.8 . The instrument is switched on and it fails to operate. If the probability t …

The probability that a randomly selected 2-digit number belongs to the set $\{n\in N:\left(2^n-2\right)$ is a multiple of 3$\}$ is equal to :

Two dices 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:

Let there be three independent events ${\mathrm{E}}_1,{\mathrm{E}}_2$ and ${\mathrm{E}}_3$. The probability that only ${\mathrm{E}}_1$ occurs is $\alpha$, only $E_2$ occurs is $\beta$ and only $E_3$ occurs is $\gamma$. Let ' $p$ ' denote the probability of none of events occ …

Let 9 distinct balls be distributed among 4 boxes, ${\mathrm{B}}_1,{\mathrm{B}}_2,{\mathrm{B}}_3$ and ${\mathrm{B}}_4$. If the probability that $B_3$ contains exactly 3 balls is $k{\left(\frac{3}{4}\right)}^9$ then $k$ lies in the set :

Let $X$ be a random variable such that the probability function of a distribution is given by $P(X=0)=\frac{1}{2},P(X=j)=\frac{1}{3^j}(j=1,2,3,\dots ,\infty )$. Then the mean of the distribution and $P(X$ is positive and even) respectively are :

A fair coin is tossed n -times such that the probability of getting at least one head is at least 0.9 . Then the minimum value of $n$ is ____

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 $\frac{1}{2}$, is :

Two squares are chosen at random on a chessboard (see figure). The probability they have a side in common is:

Let X be a random variable with distribution.

If the mean of X is 2.3 and variance of X is ${\sigma }^2$, then $100{\sigma }^2$ is equal to:

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 :

Let A denote the event that a 6-digit integer formed by $0,1,2,3,4,5,6$ without repetitions, by divisible by 3. Then probability of event A is equal to :

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 :

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 $\frac{1}{2}$ and probability of occurrence of 0 at the odd place be $\frac{1}{3}$. Then the probability that '10' is followed by 01 is …

An unbiased coin is tossed 5 times. Suppose that a variable $X$ is assigned the value $k$ when $k$ consecutive heads are obtained for $k=3,4,5$, otherwise $X$ takes the value -1. Then the expected value of $X$, is

In a workshop, there are five machines and the probability of any one of them to be out of service on a day is $\frac{1}{4}$. If the probability that at most two machines will be out of service on the same day is ${\left(\frac{3}{4}\right)}^{3}k$, then k is equal to:

Let A and B be two independent events such that $\mathrm{P}(\mathrm{A})=\frac{1}{3}$ and $\mathrm{P}(\mathrm{B})=\frac{1}{6}$. Then, which of the following is TRUE?

Let A and B be two events such that the probability that exactly one of them occurs is $\frac{2}{5}$ and the probability that A or B occurs is $\frac{1}{2}$, then the probability of both of them occur together is:

A random variable $X$ has the following probability distribution:

\begin{tabular}{ccccccc}
$X$ & $:$ & 1 & 2 & 3 & 4 & 5 \\
$P(X)$ & $:$ & $K^2$ & $2K$ & $K$ & $2K$ & $5K^2$
\end{tabular}

Then $\mathrm{P}(\mathrm{X}>2)$ is equal to:

In a box, there are 20 cards, out of which 10 are labeled 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 - ca …