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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 \( x \) 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: