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Bag \( 1 \) contains \( 4 \) white balls and \( 5 \) black balls, and Bag \( 2 \) contains \( n \) white balls and \( 3 \) black balls. One ball is drawn randomly from Bag \( 1 \) and transferred to Bag \( 2 \). A ball is then drawn randomly from Bag \( 2 \). If the probability, that the ball drawn is white, is \(\displaystyle \frac{29}{45} \), then \( n \) is equal to:

Three distinct numbers are selected randomly from the set \( (1, 2, 3, \ldots, 40) \). If the probability, that the selected numbers are in an increasing G.P., \( \frac{m}{n} \), \( \gcd(m,n)=1 \), then \( m+n \) is equal to __________.

The probability, of forming a 12 persons committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor, is:

A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let $X$ denote the number of defective pens. Then the variance of $X$ is:

A bag contains \( 19 \) unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and head turns up. If the probability that the drawn coin was unbiased, is \( \frac{m}{n} \), \( \gcd(m,n)=1 \), then \( n^{2}-m^{2} \) is equal to:

Two balls are selected at random one by one without replacement from a bag containing \( 4 \) white and \( 6 \) black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is \( \frac{m}{n} \), where \( \gcd(m,n) = 1 \), then \( m + n \) is equal to:

All five letter words are made using all the letters A, B, C, D, E and arranged as in an English dictionary with serial numbers. Let the word at serial number \( n \) be denoted by \( W_{n} \). Let the probability \( P(W_{n}) \) of choosing the word \( W_{n} \) satisfy \( P(W_{n}) = 2P(W_{n-1}), \ n > 1 \). If \( P(CDBEA) = \frac{2^{\alpha}}{2^{\beta} - 1}, \ \alpha, \beta \in \)\(\mathbb{N}\), then \( \alpha + \beta \) is equal to _________.

If \( A \) and \( B \) are two events such that \( P(A \cap B)=0.1 \), and \( P(A|B) \) and \( P(B|A) \) are the roots of the equation \( 12x^{2}-7x+1=0 \), then the value of \(\displaystyle \frac{P(\bar{A}\cup \bar{B})}{P(\bar{A}\cap \bar{B})} \) is:

A board has \( 16 \) squares as shown in the figure:

Out of these \( 16 \) squares, two squares are chosen at random. The probability that they have no side in common is:

If \( A \) and \( B \) are two events such that \( P(A) = 0.7 \), \( P(B) = 0.4 \) and \( P(A \cap \overline{B}) = 0.5 \), where \( \overline{B} \) denotes the complement of \( B \), then \( P\big(B \mid (A \cup \overline{B})\big) \) is equal to:

\( A \) and \( B \) alternately throw a pair of dice. \( A \) wins if he throws a sum of \( 5 \) before \( B \) throws a sum of \( 8 \), and \( B \) wins if he throws a sum of \( 8 \) before \( A \) throws a sum of \( 5 \). The probability, that \( A \) wins if \( A \) makes the first throw, is:

Let a random variable \( X \) take values \( 0, 1, 2, 3 \) with \( P(X=0)=P(X=1)=p \), \( P(X=2)=P(X=3)= \) and \( E(X^{2})=2E(X) \). Then the value of \( 8p-1 \) is:

If the probability that the random variable \( X \) takes the value \( x \) is given by \( P(X=x)=k(x+1)3^{-x} \), \( x = 0, 1, 2, 3, \ldots \) where \( k \) is a constant, then \( P(X \ge 3) \) is equal to:

Bag \( B_1 \) contains \( 6 \) white and \( 4 \) blue balls, Bag \( B_2 \) contains \( 4 \) white and \( 6 \) blue balls, and Bag \( B_3 \) contains \( 5 \) white and \( 5 \) blue balls. One of the bags is select at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag \( B_2 \), is:

One die has two faces marked \( 1 \), two faces marked \( 2 \), one face marked \( 3 \) and one face marked \( 4 \). Another die has one face marked \( 1 \), two faces marked \( 2 \), two faces marked \( 3 \) and one face marked \( 4 \). The probability of getting the sum of numbers to be \( 4 \) or \( 5 \), when both the dice are thrown together, is:

Let \( A = [a_{ij}] \) be a square matrix of order \( 2 \) with entries either \( 0 \) or \( 1 \). Let \( E \) be the event that \( A \) is an invertible matrix. Then the probability \( P(E) \) is:

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JEE Main 2025 7th April Evening Shift

Let a random variable \( X \) take values \( 0, 1, 2, 3 \) with \( P(X=0)=P(X=1)=p \), \( P(X=2)=P(X=3)= \) and \( E(X^{2})=2E(X) \). Then the value of \( 8p-1 \) is: