A team of 8 couples, (husband and wife) attend a lucky draw in which four persons are picked for a prize. Then the probability that there is at least one couple is

a)

b)

c)

d)

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)

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