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A bag contains $10$ balls out of which $k$ are red and $(10 - k)$ are black, where $0 \leq k \leq 10$. If three balls are drawn at random without replacement and all of them are found to be black, then the probability that the bag contains $1$ red and $9$ black balls is:

Two distinct numbers \( a \) and \( b \) are selected at random from \( 1, 2, 3, \ldots, 50 \). The probability, that their product \( ab \) is divisible by \( 3 \), is:

A candidate has to go to the examination centre to appear in an examination. The candidate uses only one means of transportation for the entire distance out of bus, scooter and car. The probabilities of the candidate going by bus, scooter and car, respectively, are \( \frac{2}{5} \), \( \frac{1}{5} \) and \( \frac{2}{5} \). The probabilities that the candidate reaches late at the examination centre are \( \frac{1}{5} \), \( \frac{1}{3} \) and \( \frac{1}{4} \) if the candidate uses bus, scooter and car, respectively. Given that the candidate reached late at the examination centre, the probability that the candidate travelled by bus is:

The probabilities that players \( A \) and \( B \) of a team are selected for the captaincy for a tournament are \( 0.6 \) and \( 0.4 \), respectively. If \( A \) is selected the captain, the probability that the team wins the tournament is \( 0.8 \) and if \( B \) is selected the captain, the probability that the team wins the tournament is \( 0.7 \). Then the probability that the team wins the tournament is:

If a random variable \( x \) has the probability distribution

Then \( P(3 < x \le 6) \) is equal to:

From a month of \( 31 \) days, \( 3 \) different dates are selected at random. If the probability that these dates are in an increasing A.P. is equal to \( \dfrac{a}{b} \) where \( a, b \in \mathbb{N} \) and \( \gcd(a,b) = 1 \) then \( a + b \) is equal to ______.

A random variable \( X \) takes values \( 0, 1, 2, 3 \) with probabilities \( \frac{2a+1}{30}, \frac{8a-1}{30}, \frac{4a+1}{30}, b \) respectively, where \( a, b \in \mathbb{R} \). Let \( \mu \) and \( \sigma \) respectively be the mean and standard deviation of \( X \) such that \( \sigma^2 + \mu^2 = 2 \). Then \( \frac{a}{b} \) is equal to:

A bag contains \( 6 \) blue and \( 6 \) green balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each drawn pair consists of one blue and one green ball is:

Let \( S \) be a set of \( 5 \) elements and \( P(S) \) denote the power set of \( S \). Let \( E \) be an event of choosing an ordered pair \( (A,B) \) from the set \( P(S)\times P(S) \) such that \( A\cap B=\phi \). If the probability of the event \( E \) is \( \dfrac{3^{p}}{2^{q}} \), where \( p,q\in\mathbb{N} \), then \( p+q \) is equal to ____________ .

A letter is known to have arrived by post either from KANPUR or from ANANTPUR. On the envelope just two consecutive letters AN are visible. The probability, that the letter came from ANANTPUR, is:

A bag contains \( (N + 1) \) coins – \( N \) fair coins, and one coin with ‘Head’ on both sides. A coin is selected at random and tossed. If the probability of getting ‘Head’ is \( \frac{9}{16} \), then \( N \) is equal to:

Bag A contains \( 9 \) white and \( 8 \) black balls, while bag B contains \( 6 \) white and \( 4 \) black balls. One ball is randomly picked up from the bag B and mixed up with the balls in the bag A. Then a ball is randomly drawn from the bag A. If the probability, that the ball drawn is white, is \( \dfrac{p}{q} \), \( \gcd(p,q)=1 \), then \( p+q \) is equal to:

A man throws a fair coin repeatedly. He gets \( 10 \) points for each head he throws and \( 5 \) points for each tail he throws. If the probability that he gets exactly \( 30 \) points is \( \frac{m}{n} \), \( \gcd(m,n)=1 \), then \( m+n \) is equal to:

From the first \( 100 \) natural numbers, two numbers first \( a \) and then \( b \) are selected randomly without replacement. If the probability that \( a - b \ge 10 \) is \( \frac{m}{n} \), \( \gcd(m, n) = 1 \), then \( m + n \) is equal to \( \_\_\_\_\_ \).

From a lot containing \( 10 \) defective and \( 90 \) non-defective bulbs, \( 8 \) bulbs are selected one by one with replacement. Then the probability of getting at least \( 7 \) defective bulbs is:

Let \( a,b,c \in \{1,2,3,4\} \). If the probability, that \( ax^{2}+2\sqrt{2}bx+c>0 \) for all \( x\in \mathbb{R} \), is \( \frac{m}{n} \), \( \gcd(m,n)=1 \), then \( m+n \) is equal to ______.

A coin is tossed \( 8 \) times. If the probability that exactly \( 4 \) heads appear in the first six tosses and exactly \( 3 \) heads appear in the last five tosses is \( p \), then \( 96p \) is equal to ________.

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JEE Main 2026 2nd April Morning Shift

Let \( a,b,c \in \{1,2,3,4\} \). If the probability, that \( ax^{2}+2\sqrt{2}bx+c>0 \) for all \( x\in \mathbb{R} \), is \( \frac{m}{n} \), \( \gcd(m,n)=1 \), then \( m+n \) is equal to ______.