Probability

Question 11

Suppose we uniformly and randomly select a permutation from the 20! Permutations of 1, 2, 3 ,…..,20. What is the probability that 2 appears at an earlier position than any other even number in the selected permutation?

• 1/2
• 1/10
• 9!/20!
• Node of the above

Question 12

We are given a set X = {x1, .... xn} where xi = 2ⁱ. A sample S ⊆ X is drawn by selecting each xi independently with probability pi = 1/2. The expected value of the smallest number in sample S is:

• 1/n
• 2
• sqrt(n)
• n

Question 13

Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is ________

• 0.24 to 0.27
• 0.15 to 0.30
• 0.20 to 0.30
• 0.10 to 0.15

Question 14

Four fair six-sided dice are rolled. The probability that the sum being 22 is X/1296. The value of X is ________

• 7
• 8
• 9
• 10

Question 15

The security system at an IT office is composed of 10 computers of which exactly four are working. To check whether the system is functional, the officials inspect four of the computers picked at random (without replacement). The system is deemed functional if at least three of the four computers inspected are working. Let the probability that the system is deemed functional be denoted by p. Then 100p = _____________.

• 11.85

Question 16

Each of the nine words in the sentence ”The quick brown fox jumps over the lazy dog” is written on a separate piece of paper. These nine pieces of paper are kept in a box. One of the pieces is drawn at random from the box. The expected length of the word drawn is _____________. (The answer should be rounded to one decimal place.)

• 3.8 to 3.9

Question 17

Let S be the sample space and two mutually exclusive events A and B be such that A U B = S. If P(.) denotes the probability of the event. The maximum value of P(A)P(B) is ______

• 0.5
• 0.25
• 0.225
• 0.125

Question 18

For each element in a set of size 2n, an unbiased coin is tossed. The 2n coin tosses are independent. An element is chosen if the corresponding coin toss were head. The probability that exactly n elements are chosen is:

• (2nCn) / (4^n)
• (2nCn) / (2^n)
• 1 / (2nCn)
• 1/2

Question 19

Let f(x) be the continuous probability density func­tion of a random variable X. The probability that a < X ≤ b, is

		 
A) f(b - a)
B) f(b) - f(a)
C)  
D) 

• A
• B
• C
• D

Question 20

Box P has 2 red balls and 3 blue balls and box Q has 3 red balls and 1 blue ball. A ball is selected as follows:

(i)  Select a box
(ii) Choose a ball from the selected box such that each ball in
     the box is equally likely to be chosen. The probabilities of
     selecting boxes P and Q are (1/3) and (2/3), respectively.  

Given that a ball selected in the above process is a red ball, the probability that it came from the box P is

• 4/19
• 5/19
• 2/9
• 19/30