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A bag contains \( 4 \) white and \( 6 \) black balls. Three balls are drawn at random from the bag. Let \( X \) be the number of white balls, among the drawn balls. If \( \sigma^{2} \) is the variance of \( X \), then \( 100\sigma^{2} \) is equal to _______________.

Let \( S = \{E1, E2 \ldots, E8\} \) be a sample space of a random experiment such that \( P(En) = \frac{n}{36} \) for every \( n = 1, 2, \ldots, 8 \). Then the number of elements in the set \( \{ A \subseteq S : P(A) \ge \frac{4}{5} \} \) is _______.

Let \( S \) be the sample space of all five digit numbers. If \( p \) is the probability that a randomly selected number from \( S \), is a multiple of \( 7 \) but not divisible by \( 5 \), then \( 9p \) is equal to:

Let \( E_{1} \) and \( E_{2} \) be two events such that the conditional probabilities \( P(E_{1}\mid E_{2}) = \frac{1}{2} \), \( P(E_{2}\mid E_{1}) = \frac{3}{4} \) and \( P(E_{1}\cap E_{2}) = \frac{1}{8} \). Then:

A six faced die is biased such that \( 3\times P(\text{a prime number})=6\times P(\text{a composite number})=2\times P(1) \). Let \( X \) be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the mean of \( X \) 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 _________.

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 at most two heads is:

If the sum and the product of mean and variance of a binomial distribution are \( 24 \) and \( 128 \) respectively, then the probability of one or two successes 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 \( n \) is equal to ________.

Let \( X \) have a binomial distribution \( B(n,p) \) such that sum and the product of the mean and variance of \( X \) are \( 24 \) and \( 128 \) respectively. If \( P(X>n-3)=\dfrac{k}{2^n} \), then \( k \) is equal to:

Let A and B two events such that \( P(B \mid A) = \dfrac{2}{5} \), \( P(A \mid B) = \dfrac{1}{7} \) and \( P(A \cap B) = \dfrac{1}{9} \). Consider:
(S1) \( P(A \cup B) = \dfrac{5}{6} \), (S2) \( P(A' \cup B') = \dfrac{1}{18} \).
Then:

Out of \( 60\% \) female and \( 40\% \) male candidates appearing in an exam, \( 60\% \) candidates qualify it. The number of females qualifying the exam is twice the number of males qualifying it. A candidate is randomly chosen from the qualified candidates. The probability, that the chosen candidate is a female, is:

Bag \( I \) contains \( 3 \) red, \( 4 \) black and \( 3 \) white balls and Bag \( II \) contains \( 2 \) red, \( 5 \) black and \( 2 \) white balls. One ball is transferred from Bag \( I \) to Bag \( II \) and then a ball is drawn from Bag \( II \). The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is:

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 \( E_1, E_2, E_3 \) be three mutually exclusive events such that \( P(E_1) = \dfrac{3+3p}{6} ,\ P(E_2) = \dfrac{2-p}{8} \) and \( P(E_3) = \dfrac{1-p}{2} \). If the maximum and minimum values of \( p \) are \( p_1 \) and \( p_2 \), then \( (p_1 + p_2) \) is equal to:

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

If \( A \) and \( B \) are two events such that \( P(A)=\dfrac{1}{3} \), \( P(B)=\dfrac{1}{5} \) and \( P(A \cup B)=\dfrac{1}{2} \) then \( P(A \mid B') + P(B \mid A') \) 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 answers of exactly \( 8 \) questions correctly out of \( 10 \) is \( \frac{27k}{4^{10}} \), then \( k \) is equal to ________ .

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

Five numbers \( x_1, x_2, x_3, x_4, x_5 \) are randomly selected from the numbers \( 1,2,3,\ldots,18 \) and are arranged in the increasing order \( (x_1 < x_2 < x_3 < x_4 < x_5) \). The probability that \( x_2 = 7 \) and \( x_4 = 11 \) is:

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

If the numbers appeared on the two thrown of a fair six faced die are \( \alpha \) & \( \beta \), then the probability that \( x^{2}+\alpha x+\beta>0 \), for all \( x\in \mathbb{R} \) is:

A random variable \( X \) has the following probability distribution:

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

The probability, that in a randomly selected \( 3 \)-digit number at least \( 2 \) digits are odd, is:

If a random variable \( X \) follows the Binomial distribution \( B(33,\ p) \) such that \( 3P(X=0) = P(X=1) \), then the value of \( \dfrac{P(X=15)}{P(X=18)} - \dfrac{P(X=16)}{P(X=17)} \) is equal to:

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JEE Main 2022 25th June Morning Shift

Let \( E_{1} \) and \( E_{2} \) be two events such that the conditional probabilities \( P(E_{1}\mid E_{2}) = \frac{1}{2} \), \( P(E_{2}\mid E_{1}) = \frac{3}{4} \) and \( P(E_{1}\cap E_{2}) = \frac{1}{8} \). Then: