Four persons can hit a target correctly with probabilities {1 \over 2}, {1 \over 3}, {1 \over 4} and {1 \over 8} respectively. if all hit at the target independently, then the probability that the target would be hit, is :

The minimum number of times one has to toss a fair coin so that the probability of observing at least one head is at least 90% is :

Let A and B be two non-null events such that A $$\subset$$ B . Then, which of the following statements is always correct?

In a game, a man wins Rs. 100 if he gets 5 or 6 on a throw of a fair die and loses Rs. 50 for getting any other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum of three throws, then his expected gain/loss (in rupees) is :

In a class of 60 students, 40 opted for NCC, 30 opted for NSS and 20 opted for both NCC and NSS. If one of these students is selected at random, then the probability that the students selected has opted neither for NCC nor for NSS is :

An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn; the original ball is not returned to the urn. Now, a second ball is drawn at random from it. The probability that the second ball is red, is :

An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered 1, 2, 3, ……, 9 is randomly picked and the number on the card is noted. The probability that the noted number is either 7 or 8 is :

If the probability of hitting a target by a shooter, in any shot, is {1 \over 3}, then the minimum number of independent shots at the target required by him so that the probability of hitting the target atleast once is greater than {5 \over 6} is :

Two integers are selected at random from the set {1, 2, ...., 11}. Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is :

Let S = {1, 2, . . . . . ., 20}. A subset B of S is said to be "nice", if the sum of the elements of B is 203. Then the probability that a randonly chosen subset of S is "nice" is :

A bag contains 30 white balls and 10 red balls. 16 balls are drawn one by one randomly from the bag with replacement. If X be the number of white balls drawn, then \left( {{{mean\,\,of\,X} \over {s\tan dard\,\,deviation\,\,of\,X}}} \right) is equal to :

In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to :

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the random variable of number of aces obtained in the two drawn cards. Then P(X = 1) + P (X = 2) equals :

The derivative of {\tan ^{ - 1}}\left( {{{\sin x - \cos x} \over {\sin x + \cos x}}} \right), with respect to {x \over 2} , where \left( {x \in \left( {0,{\pi \over 2}} \right)} \right) is :

If ey + xy = e, the ordered pair \left( {{{dy} \over {dx}},{{{d^2}y} \over {d{x^2}}}} \right) at x = 0 is equal to :

Let f(x) = loge(sin x), (0 < x < $$\pi$$) and g(x) = sin–1 (e–x ), (x $$\ge$$ 0). If $$\alpha$$ is a positive real number such that a = (fog)'($$\alpha$$) and b = (fog)($$\alpha$$), then :

If ƒ(1) = 1, ƒ'(1) = 3, then the derivative of ƒ(ƒ(ƒ(x))) + (ƒ(x))2 at x = 1 is :

If 2y = {\left( {{{\cot }^{ - 1}}\left( {{{\sqrt 3 \cos x + \sin x} \over {\cos x - \sqrt 3 \sin x}}} \right)} \right)^2}, x $$\in$$ \left( {0,{\pi \over 2}} \right) then $$dy\frac{d}{x}$$ is equal to:

If xloge(logex) $$-$$ x2 + y2 = 4(y > 0), then {{dy} \over {dx}} at x = e is equal to :

Let f : R $$\to$$ R be a function such that f(x) = x3 + x2f'(1) + xf''(2) + f'''(3), x $$\in$$ R. Then f(2) equals -