A₁, A₂, .  .  .  ., A_n are n independent events with  (1 ≤ j ≤ n). The probability that not one of A₁, A₂, .  .  .  ., A_n occur is

a)

b)

c)

d) None of these

A and B are two events. Odds against A are 2 : 1. Odds in favour of A ⋃ B are 3 : 1. If x ≤ P(B) ≤ y, then the ordered pair (x, y) is

a)

b)

c)

d) None of the above

If X is a binomial variate with parameters n and p, where 0 < p < 1 such that  is independent of n and r, then p equals

a)

b)

c)

d) None of these

Let X denote the number of times heads occur in n tosses of a fair coin. If P (X = 4), P (X = 5) and P (X = 6) are in arithmetic progression then the value of n is

a) 7, 14

b) 10

c) 12

d) None of these

The least number of times a fair coin must be tossed so that the probability of getting at least one head is at least 0.8 is

a) 7

b) 6

c) 5

d) 3

A biased coin with probability p, 0 < p < 1 of heads is tossed until a head appears for the first time. If the probability that the number of tosses required is even is , then p equals

a)

b)

c)

d)

A bag contains n + 1 coins. It is known that one of these coins shows heads on both the sides, whereas the other coins are fair. One coin is selected at random and tossed. If the probability that the toss results in head is  then the value of n is

a) 3

b) 4

c) 5

d) None of these

In the following question each question contains Statement 1 (Assertion) and Statement 2 (Reason). Each question has four choices (a), (b), (c), (d) out of which only one is correct.
Statement 1: Out of 5 tickets consecutively numbered, three are drawn at random, the chance that the numbers on them are in arithmetic progression is
Statement 2: Out of (2n + 1) tickets consecutively numbered, three are drawn at random, the chance that the numbers on them are in arithmetic progression is

a) Statement 1 is true. Statement 2 is true. Statement 2 is a correct explanation for statement 1.

b) Statement 1 is true. Statement 2 is true. Statement 2 is NOT a correct explanation for statement 1.

c) Statement 1 is true. Statement 2 is false.

d) Statement 1 is false. Statement 2 is true

A die is thrown 3 times. What is the probability that an odd number turns up
i)Exactly 4 times
ii)At least 4 times

If m things are distributed among ‘a’ men and ‘b’ women, show that the probability that the number of things received by men is odd, is  is

In a box there are 2 red, 3 black and 4 white balls. Out of these three balls are drawn together. The probability of these balls being of the same colour is

a) 1/84

b) 1/21

c) 5/84

d) None of these

In a hurdle race, a runner has probability p of jumping over a specific hurdle. Given that in 5 trials, the runner succeeded 3 times, the conditional probability that the runner has succeeded in the first trial is

a) 3/5

b) 2/5

c) 1/5

d) None of these

Let E and F be two independent events. The probability that both E and F happens is 1/12 and the probability that neither E nor F happens is ½. Then

a) P (E) = 1/3, P (F) = 1/4 or P (E) = 1/4, P (F) = 1/3

b) P (E) = 1/2, P (F) = 1/6

c) P (E) = 1/6, P (F) = 1/2

d) P (E) = 1/4, P (F) = 1/3

Consider the experiment of distribution of balls among urns. Suppose we are given M urns, numbered 1 to M, among which we are to distribute n balls (n < M). Let P (A) denote the probability that each of the urns numbered 1 to n will contain exactly one ball. If the balls are different and any ball can go to any urn, then P (A) is equal to

a)

b)

c)

d)

Consider the experiment of distribution of balls among urns. Suppose we are given M urns, numbered 1 to M, among which we are to distribute n balls (n < M). Let P (A) denote the probability that each of the urns numbered 1 to n will contain exactly one ball. If the balls are identical but at most one ball can be put in any box, then P (A) is equal to

a)

b)

c)

d)

An artillery target may be either at point A with probability  or at point B with probability . We have 21 shells each of which can be targeted either at point A or point B. Each shell may hit the target independently of the other shell with probability . If the number of shells that must be fired at point A to hit the target with maximum probability is λ then   is

If the minimum number of tosses of a pair of dice, so that the probability of getting the sum of the numbers on the dice equal to 7 on at least one toss, is greater than 0.95, is λ, then λ is

(Given log₁₀2 = 0.3010, log₁₀3 = 0.4771)

Two teams A and B play a tournament. The first one to win (n + 1) games wins the series. The probability that A wins a game is p and that B wins a game is q (no ties). If the value of , then λ is

A bag contains a large number of white and black marbles in equal proportions. Two samples of 5 marbles are selected (with replacement) at random. The probability that the first sample contains exactly one black marble and the second sample contains exactly 3 black marbles, is , then a + b is

If two events A and B are such that P(A’) = 0.3, P (B) = 0.4 and P (A ⋂ B’) = 0.5. If  then a + b is

A four digit number is formed using the digits 0, 1, 2, 3, 4 without repetition. If the probability that it is divisible by 4 is  then b – 2a is

In throwing of a die, let A be the event ‘an odd number turns up’, B be the event ‘a number divisible by three turns up’ and C be the event ‘a number ≤ 4 turns up’. The probability that exactly two of A, B and C occur is  , then a + b is

Two switches S₁ and S₂ have respectively 90% and 80% chances of working. If the probability that the following circuit will work =  , then b – a is