Cards of an ordinary deck of playing cards are placed into two heaps. Heap 1 consists of all the red cards and Heap 2 consists of all the black cards. A heap is chosen at random and a card is drawn. If the probability that the card drawn is a king =  then b – 4a is

A pack of cards is counted with face downwards. It is found that one card is missing. One card is drawn and is found to be red. If the probability that the missing card is red is  then b – 2a is

Let ω be a complex cube root of unity with ω ≠ 1. A fair die is thrown three times. If r₁, r₂ and r₃ are the numbers obtained on the die, then the probability that  is

a)

b)

c)

d)

A signal which can be green or red with probability  and  respectively, is received by station A and then transmitted to station B. The probability of each station receiving the signal correctly is . If the signal received at station B is green, then the probability that the original signal is green is

a)

b)

c)

d)

A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required. The probability that X = 3 equals

a)

b)

c)

d)

A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required. The probability that X ≥ 3 equals

a)

b)

c)

d)

A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required. The conditional probability that X ≥ 6 given X > 3 equals

a)

b)

c)

d)

Consider the system of equations
ax + by = 0, cx + dy = 0, where a, b, c, d  {0, 1}

Statement 1 – The probability that the system of equations has a unique solution is 3/8

Statement 2 – The probability that the system of equations has a solution is 1.

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

An experiment has 10 equally likely outcomes. Let A and B be two non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent is

a) 2, 4 or 8

b) 3, 6 or 9

c) 4 or 8

d) 5 or 10

Let E^c denote the complement of an event E. Let E, F, G be pairwise independent events with P (G) > 0 and P (E ⋂ F ⋂ G) = 0 Then P (E^c ⋂ F^c | G) equals

a) P (E^c) + P (F^c)

b) P (E^c) − P (F^c)

c) P (E^c) − P (F)

d) P (E) − P (F^c)

There are n urns each containing (n+1) balls such that the i^th urn contains “i” white balls and (n+1−i) red balls. Let u_i be the event of selecting the i^th urn, i = 1, 2, 3 ..., n, and w denote the event of getting a white ball. If P (u_i) α i , where i = 1, 2, 3,.  .  ., n, the  is equal to

a) 1

b)

c)

d)

There are n urns each containing (n+1) balls such that the i^th urn contains “i” white balls and (n+1−i) red balls. Let u_i be the event of selecting the i^th urn, i = 1, 2, 3 ..., n, and w denote the event of getting a white ball. If P (u_i) = c, where c is constant, then
 is equal to

a)

b)

c)

d)

There are n urns each containing (n+1) balls such that the i^th urn contains “i” white balls and (n+1−i) red balls. Let u_i be the event of selecting the i^th urn, i = 1, 2, 3 ..., n, and w denote the event of getting a white ball. If n is even and E denotes the event of choosing an even numbered urn {P (u_i) =  }, then the value of  is equal to

a)

b)

c)

d)

An urn contains 3 red and 5 blue balls. The probability that two balls are drawn in which the second ball drawn is blue without replacement is

a)

b)

c)

d)

If A, B, C are three events, then show that
P (exactly two of A, B, C occur) ≤ P (A ⋂ B) + P (B ⋂ C) + P (C ⋂ A)

If A, B, C are three events, then show that
P (A ⋃ B ⋃ C) ≤ P (A) + P (B) + P (C)

If A, B, C are three events, then show that
P (exactly one  of A, B, C occur) ≤ P (A) + P (B) + P (C)− P (B ⋂ C) − P (C ⋂ A) − P (A ⋂ B)

If A, B, C are three events, then show that
P (A and at least one of B, C occurs ) ≤ P (A ⋂ B) + P (A ⋂ C)

If
 is equal to  then find the value of p.

Let A, B , C be three mutually independent events. Prove that A and B ∪ C are independent.

Let A, B , C be three mutually independent events. Prove that A and B ∩ C are independent.

Prove that for any two events in the sample space
 

For any two events in the sample space show