Q16: Consider a dice with the property that the probability of a face with n dots showing up proportional to n. The probability of the face with three dots showing up is____. (SET-2 (2014))
(a) 0.1
(b) 0.33
(c) 0.14
(d) 0.66
Ans: (c)
Sol: Let probability of occurence of one dot is P.
So, writing total probability
P + 2P + 3P + 4P + 5P + 6P = 1
P = 1/21
Hence, problem of occurrence of 3 dot is = 3P = 3/21 = 1/7 = 0.142
Q17: A fair coin is tossed n times. The probability that the difference between the number of heads and tails is (n-3) is (SET-1 (2014))
(a) 2-n
(b) 0
(c) 
(d) 2-n+3
Ans: (b)
Sol: 
when, n = probability of occurrence of head
y = probability of occurrence of tail
Let number of head is P.
Number of tail is q
P + q = n
Total number of tails
Given: |P - q| = n - 3
|P-(n-P)| = n - 3
n = n - 3 which is not posssible.
Here required probability is zero.
Q18: A continuous random variable X has a probability density function f(x) = e−x, 0 < x < ∞. Then P{X > 1} is (2013)
(a) 0.368
(b) 0.5
(c) 0.632
(d) 1
Ans: (a)
Sol: 
Q19: A fair coin is tossed till a head appears for the first time. The probability that the number of required tosses is odd, is (2012)
(a) 1/3
(b) 1/2
(c) 2/3
(d) 3/4
Ans: (c)
Sol: P(number of tosses is odd) = P(number of tosses is 1, 3, 5, 7?)
P(no. of toss is 1) = P(Head in first toss) = 1/2
P(no. of toss is 3) = P( tail in first toss , tail in second toss and head in third toss) 
P(no. of toss id 5) = P(T, T, T, T, H) = (1/2)5 = 1/32
So P(no. of tosses is odd) 
Sum of infinite geometric series with a = 1/2 and r = 1/4
Q20: Two independent random variables X and Y are uniformly distributed in the interval [-1, 1]. The probability that max [X, Y] is less than 1/2 is (2012)
(a) 3/4
(b) 9/16
(c) 1/4
(d) 2/3
Ans: (b)
Sol: −1 ≤ x ≤ 1 and −1 ≤ y ≤ 1 is the entire rectangle.The region in which maximum of {x, y} is less than 1/2 is shown below as shaded regioninside this rectangle.
Q21: A box contains 4 white balls and 3 red balls. In succession, two balls are randomly and removed form the box. Given that the first removed ball is white, the probability that the second removed ball is red is (2010)
(a) 1/3
(b) 3/7
(c) 1/2
(d) 4/7
Ans: (c)
Sol: 
Q22: Assume for simplicity that N people, all born in April (a month of 30 days), are collected in a room. Consider the event of at least two people in the room being born on the same date of the month, even if in different years, e.g. 1980 and 1985. What is the smallest N so that the probability of this event exceeds 0.5 ? (2009)
(a) 20
(b) 7
(c) 15
(d) 16
Ans: (b)
Q23: X is a uniformly distributed random variable that takes values between 0 and 1. The value of E{X3} will be (2008)
(a) 0
(b) 1/8
(c) 1/4
(d) 1/2
Ans: (c)
Sol: x is uniformly distributes in [0, 1]
Therefore, probability density function
Now,
Q24: A loaded dice has following probability distribution of occurrences
If three identical dice as the above are thrown, the probability of occurrence of values 1, 5 and 6 on the three dice is (2007)
(a) same as that of occurrence of 3, 4, 5
(b) same as that of occurrence of 1, 2, 5
(c) 1/128
(d) 5/8
Ans: (c)
Sol: 
Q25: Two fair dice are rolled and the sum r of the numbers turned up is considered (2006)
(a) Pr(r > 6) = 1/6
(b) Pr (r/3 is an integer) = 5/6
(c) Pr (r = 8 | r/4 is an integer) = 5/9
(d) Pr (r = 6 |r/5 is an integer) = 1/18
Ans: (c)
Sol: If two fair dices are rolles the probability distribution of r where r is the sum of the numbers on each die is given by
The above table has been obtained by taking all different ways of obtaining a particular sum. For example, a sum of 5 can be obtained by (1, 4), (2, 3), (3, 2), and (4, 1).
∴ P(x = 5) = 4/36
Now let us consider choise (A)
Consider choise (C)
Pr(r = 8∣r/4 is an integer)= 1/36
Now,
Choice (C) is correct.
Q26: A fair coin is tossed three times in succession. If the first toss produces a head, then the probability of getting exactly two heads in three tosses is (2005)
(a) 1/8
(b) 1/2
(c) 3/8
(d) 3/4
Ans: (b)
Sol: 
Q27: If P and Q are two random events, then the following is TRUE (2005)
(a) Independence of P and Q implies that probability (P ∩ Q) = 0
(b) Probability (P ∪ Q) ≥ Probability (P) + Probability (Q)
(c) If P and Q are mutually exclusive, then they must be independent
(d) Probability (P ∩ Q) ≤ Probability (P)
Ans: (d)
Sol: (A) is false since of P and Q are independent
Pr(P ∩ Q) = Pr(P) × Pr(Q)
which need not be zero.
(B) is false since
(C) is false since independence and mutually exclusion are unrelated properties.
(D) is true
