A die is rolled so that the probability of face i is proportional to i, i = 1, 2, .  .  . 6. The probability of an even number occurring when the die is rolled is

a)

b)

c)

d) None of these

A and B draw two cards each, one after another, from a well shuffled pack of 52 cards. The probability that all four cards drawn are of the same suit is

a)

b)

c)

d) None of these

From a group of 10 persons consisting of 5 lawyers, 3 doctors and 2 engineers, four persons are selected at random. The probability that the selection contains at least one of each category is

a)

b)

c)

d) None of these

For any two independent events E₁ and E₂ in a space S,  is equal to

a)

b)

c)

d)

If A and B are independent events, such that , then prove that

If A and B are independent events, such that , then prove that

If A and B are independent events, such that , then prove that

If A and B are independent events, such that , then prove that

A pack of cards consists of 15 cards numbered 1 to 15. Three cards are drawn at random with replacement. Then, the probability of getting two odd and one even numbered card is

a)

b)

c)

d)

Of three independent events, the chance that only the first occurs is a, that only the second occurs is b and only the third occurs is c. Then, the probability of occurrence of these three events are [where x is a root of the equation (a + x) (b + x) (c + x) = x²]

a)

b)

c)

d) None of these

There are 4 white and 6 black balls in a box. In another box there are 3 white and 4 black balls. An unbiased dice is rolled. If it shows a number less than or equal to 3, then a ball is drawn from the second box. If the ball drawn is black, then the probability that the ball was drawn from the first box is

a)

b)

c)

d)

If  and  are complementary events of events E and F respectively and if 0 < P (F) < 1, then

a)

b)

c)

d)

The chances of defective screws in three boxes A, B, C are  respectively. A box is selected at random and a screw drawn from it at random is found to be defective. Then the probability that it came from box A is

a)

b)

c)

d)

A coin is tossed 7 times. The probability that he wins the toss on more occasions is

a)

b)

c)

d) None of these

An ordinary dice is rolled a certain number of times. The probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times. Then the probability of getting an odd number odd number of times is

a)

b)

c)

d) None of these

A fair coin is tossed a fixed number of times. If the probability of getting seven heads is equal to that of getting nine heads, the probability of getting two heads is

a)

b)

c)

d) None of these

A fair coin is tossed 100 times. The probability of getting tails an odd number of times is

a)

b)

c)

d) None of these

A fair die is thrown twenty times. The probability that on the tenth throw the fourth six appears is

a)

b)

c)

d) None of these

One hundred identical coins, each with probability p of showing heads are tossed once. If 0 < p < 1 and the probability of heads showing on 50 coins is equal to that of heads showing on 51 coins, the value of p is

a)

b)

c)

d) None of these

If the mean and variance of a binomial variate X are 2 and 1 respectively, then the probability that X takes a value greater than 1 is

a)

b)

c)

d)

If a  [−20, 0], then probability that the equation
16x² + 8 (a + 5)x – 7a – 5 = 0 has imaginary roots is

a)

b)

c)

d)

Two points are taken at random on the given straight line segment of length a. Then the probability for the distance between them to exceed a given length c, where 0 < c < a is

a)

b)

c)

d) None of these

Suppose f (x) = x³ + ax² + bx + c, where a, b, c are chosen respectively by throwing a dice three times. Then the probability that f (x) is strictly an increasing function is

a)

b)

c)

d)