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The coefficients \( a, b, c \) in the quadratic equation \( a x^{2} + b x + c = 0 \) are from the set \( \{1, 2, 3, 4, 5, 6\} \). If the probability of this equation having one real root bigger than the other is \( p \), then \( 216p \) equals

An integer is chosen at random from the integers \( 1, 2, 3, \ldots, 50 \). The probability that the chosen integer is a multiple of atleast one of \( 4 \), \( 6 \) and \( 7 \) is:

Bag A contains \( 3 \) white, \( 7 \) red balls and Bag B contains \( 3 \) white, \( 2 \) red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag A, if the ball drawn is white, is:

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the \( i^{\text{th}} \) roll than the number obtained in the \( (i-1)^{\text{th}} \) roll, \( i = 2, 3 \) is equal to

A fair die is thrown until \( 2 \) appears. Then the probability, that \( 2 \) appears in even number of throws, is:

Two marbles are drawn in succession from a box containing \( 10 \) red, \( 30 \) white, \( 20 \) blue and \( 15 \) orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white, is:

Three balls are drawn at random from a bag containing \( 5 \) blue and \( 4 \) yellow balls. Let the random variables \( X \) and \( Y \) respectively denote the number of blue and yellow balls. If \( \overline{X} \) and \( \overline{Y} \) are the means of \( X \) and \( Y \) respectively, then \( 7\overline{X} + 4\overline{Y} \) is equal to _______.

A company has two plants \( A \) and \( B \) to manufacture motorcycles. 60% motorcycles are manufactured at plant \( A \) and the remaining are manufactured at plant \( B \). 80% of the motorcycle manufactured at plant \( A \) are rated of the standard quality, while 90% of the motorcycles manufactured at plant \( B \) are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If \( p \) is the probability that it was manufactured at plant \( B \), then \( 126p \) is:

Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable \( x \) to be the number of rotten apples in a draw of two apples, the variance of \( x \) is:

Let the sum of two positive integers be \( 24 \). If the probability, that their product is not less than \( \frac{3}{4} \) times their greatest possible product, is \( \frac{m}{n} \), where \( \gcd(m,n) = 1 \), then \( n - m \) equals:

In a tournament, a team plays \( 10 \) matches with probabilities of winning and losing each match as \( \tfrac{1}{3} \) and \( \tfrac{2}{3} \) respectively. Let \( x \) be the number of matches that the team wins, and \( y \) be the number of matches that team loses. If the probability \( P(|x-y|\le 2) \) is p , then \( 3^{9}p \) equals __________.

There are three bags \( X \), \( Y \) and \( Z \). Bag \( X \) contains \( 5 \) one–rupee coins and \( 4 \) five-rupee coins; Bag \( Y \) contains \( 4 \) one-rupee coins and \( 5 \) five-rupee coins and Bag \( Z \) contains \( 3 \) one-rupee coins and \( 6 \) five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag \( Y \), is:

If three letters can be posted to any one of the \( 5 \) different addresses, then the probability that the three letters are posted to exactly two addresses is:

Let the mean and the standard deviation of the probability distribution

be \( \mu \) and \( \sigma \), respectively. If \( \sigma - \mu = 2 \), then \( \sigma + \mu \) is equal to \( \_\_\_\_\_\_\_ \).

An urn contains \( 6 \) white and \( 9 \) black balls. Two successive draws of \( 4 \) balls are made without replacement. The probability, that the first draw gives all white balls and the second draw gives all black balls, is:

A bag contains \( 8 \) balls, whose colours are either white or black. \( 4 \) balls are drawn at random without replacement and it was found that \( 2 \) balls are white and other \( 2 \) balls are black. The probability that the bag contains equal number of white and black balls is:

Three urns \( A \), \( B \) and \( C \) contain \( 7 \) red, \( 5 \) black; \( 5 \) red, \( 7 \) black and \( 6 \) red, \( 6 \) black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn \( A \) is:

The coefficients \( a,b,c \) in the quadratic equation \( ax^2 + bx + c = 0 \) are chosen from the set \( \{1,2,3,4,5,6,7,8\} \). The probability of this equation having repeated roots is:

Two integers \( x \) and \( y \) are chosen with replacement from the set \( \{ 0, 1, 2, 3, \ldots, 10 \} \). Then the probability that \( |x - y| > 5 \), is:

Let \( a \), \( b \) and \( c \) denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked \( 1 \), \( 2 \), \( 3 \), \( 4 \). If the probability that \( ax^2 + bx + c = 0 \) has all real roots is \( \frac{m}{n} \), \( \gcd(m,n) = 1 \), then \( m + n \) is equal to ________.

A fair die is tossed repeatedly until a six is obtained. Let \( X \) denote the number of tosses required and let \( a = P(X = 3) \), \( b = P(X \ge 3) \) and \( c = P(X \ge 6 \mid X > 3) \). Then \( \frac{b + c}{a} \) is equal to __________.

A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed \( 3 \) times, then the probability of getting two tails and one head is:

Let Ajay will not appear in JEE exam with probability \( p = \dfrac{2}{7} \), while both Ajay and Vijay will appear in the exam with probability \( q = \dfrac{1}{5} \). Then the probability, that Ajay will appear in the exam and Vijay will appear is:

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JEE Main 2024 31st Jan Morning Shift

Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable \( x \) to be the number of rotten apples in a draw of two apples, the variance of \( x \) is: