Let $$\mathrm{S}=\{w_1,w_2,......\}$$ be the sample space associated to a random experiment. Let P({w_n}) = {{P({w_{n - 1}})} \over 2},n \ge 2. Let $$A=\{2k+3l:k,l\in N\}$$ and $$B=\{w_n:n\in A\}$$. Then P(B) is equal to :

Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is :

Let N be the sum of the numbers appeared when two fair dice are rolled and let the probability that $$N-2,\sqrt{3N},N+2$$ are in geometric progression be $$\frac{k}{48}$$. Then the value of k is :

Let M be the maximum value of the product of two positive integers when their sum is 66. Let the sample space S = \left\{ {x \in \mathbb{Z}:x(66 - x) \ge {5 \over 9}M} \right\} and the event $$\mathrm{A}=\{\mathrm{x}\in \mathrm{S}:\mathrm{x}\mathrm{is}\mathrm{a}\mathrm{multiple}\mathrm{of}3\}$$. Then P(A) is equal to :

Let N denote the number that turns up when a fair die is rolled. If the probability that the system of equations $$x+y+z=1$$ $$2x+\mathrm{N}y+2z=2$$ $$3x+3y+\mathrm{N}z=3$$ has unique solution is {k \over 6}, then the sum of value of k and all possible values of N is :

Let $$\Omega$$ be the sample space and $$\mathrm{A}\subseteq \mathrm{\Omega }$$ be an event. Given below are two statements : (S1) : If P(A) = 0, then A = $$\varphi$$ (S2) : If P(A) = 1, then A = $$\Omega$$ Then :

A bag contains 6 white and 4 black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The probability that all the balls drawn are white is :

The random variable $$\mathrm{X}$$ follows binomial distribution $$\mathrm{B}(\mathrm{n},\mathrm{p})$$, for which the difference of the mean and the variance is 1 . If $$2\mathrm{P}(\mathrm{X}=2)=3\mathrm{P}(\mathrm{X}=1)$$, then $$n^2\mathrm{P}(\mathrm{X}>1)$$ is equal to :

A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If $$\mathrm{X}$$ denotes the number of tosses of the coin, then the mean of $$\mathrm{X}$$ is :

Two dice A and B are rolled. Let the numbers obtained on A and B be $$\alpha$$ and $$\beta$$ respectively. If the variance of $$\alpha -\beta$$ is $$\frac{p}{q}$$, where $$p$$ and $$q$$ are co-prime, then the sum of the positive divisors of $$p$$ is equal to :

Let $$S=\left\{M=\left[a_{ij}\right],a_{ij}\in \{0,1,2\},1\le i,j\le 2\right\}$$ be a sample space and $$A=\{M\in S:M$$ is invertible $$\}$$ be an event. Then $$P(A)$$ is equal to :

Let a die be rolled $$n$$ times. Let the probability of getting odd numbers seven times be equal to the probability of getting odd numbers nine times. If the probability of getting even numbers twice is $$\frac{k}{2^{15}}$$, then $$\mathrm{k}$$ is equal to :

Let N denote the sum of the numbers obtained when two dice are rolled. If the probability that $${2^N}

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

In a bolt factory, machines $$A,B$$ and $$C$$ manufacture respectively $$20\%,30\%$$ and $$50\%$$ of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found the defective, then the probability that it is manufactured by the machine $$C$$ is :

A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at least 4 successes is $$\frac{k}{3^{11}}$$, then $$k$$ is equal to :

Three dice are rolled. If the probability of getting different numbers on the three dice is $$\frac{p}{q}$$, where $$p$$ and $$q$$ are co-prime, then $$q-p$$ is equal to :

If the solution of the equation $${\log }_{\cos x}\cot x+4{\log }_{\sin x}\tan x=1,x\in \left(0,\frac{\pi }{2}\right)$$, is $${\sin }^{-1}\left(\frac{\alpha +\sqrt{\beta }}{2}\right)$$, where $$\alpha$$, $$\beta$$ are integers, then $$\alpha +\beta$$ is equal to :

The number of integral solutions $$x$$ of $${\log }_{\left(x+\frac{7}{2}\right)}{\left(\frac{x-7}{2x-3}\right)}^2\ge 0$$ is :

If $$y(x)=x^x,x>0$$, then $$y^{\prime \prime }(2)-2y^{\prime }(2)$$ is equal to