A pot contains two red balls and two blue balls. Two balls are drawn from this pot randomly without replacement.

What is the probability that the two balls drawn have different colours?

Two fair dice (with faces labeled , and 6) are rolled. Let the random variable  denote the sum of the outcomes obtained.

The expectation of  is _________ (rounded off to two decimal places).

The bar graph shows the frequency of the number of wickets taken in a match by a bowler in her career. For example, in 17 of her matches, the bowler has taken 5 wickets each. The median number of wickets taken by the bowler in a match is __________ (rounded off to one decimal place).

A state transition diagram with states , and  and transition probabilities  is shown in the figure (e.g.,  denotes the probability of transition from state A to B ). For this state diagram, select the statement(s) which is/are universally true.

Two continuous random variables  and  are related as

Let  and  denote the variance of  and , respectively. The variances are related as

A box contains the following three coins,

I. A fair coin with head on one face and tail on the other face.

II. A coin with heads on both the faces.

III. A coin with tails on both the faces.

A coin is picked randomly from the box and tossed. Out of the two remaining coins in the box, one coin is then picked randomly and tossed. If the first toss results in a head, the probability of getting a head in the second toss is

In a high school having equal number of boy students and girl students,  of the students study Science and the remaining  students study Commerce. Commerce students are two times more likely to be a boy than are Science students. The amount of information gained in knowing that a randomly selected girl student studies Commerce (rounded off to three decimal places) is ________ bits.

The two sides of a fair coin are labelled as 0 and 1. The coin is tossed two times independently. Let M and N denote the labels corresponding to the outcomes of those tosses. For a random variable X, defined as , the expected value  (rounded off to two decimal places) is ________

 is a random variable with uniform probability density function in the interval . For , the conditional probability   (rounded off to three decimal places) is __________.

Three fair cubical dice are thrown simultaneously. The probability that all three dice have the same number of dots on the faces showing up is (up to third decimal place) _______

Passengers try repeatedly to get a seat reservation in any train running between two stations until they are successful. If there is 40% chance of getting reservation in any attempt by a passenger, then the average number of attempts that passengers need to make to get a seat reserved is

Two random variables X and Y are distributed according to

The probability  is ___________.        

The probability of getting a “head” in a single toss of a biased coin is 0.3. The coin is tossed repeatedly till a “head” is obtained. If the tosses are independent, then the probability of getting “head” for the first time in the fifth toss is __

Suppose A and B are two independent events with probabilities P(A)≠0 and P(B)≠ 0. Let  and  be their complements. Which one of the following statements is FALSE?

In a housing society, half of the families have a single child per family, while the remaining half have two children per family. The probability that a child picked at random, has a sibling is __________        

Let X be a real-valued random variable with E[X] and  denoting the mean values of X and , respectively. The relation which always holds true is        

 Let X be a random variable which is uniformly chosen from the set of positive odd numbers less than 100. The expectation, E[X], is _

An unbiased coin is tossed an infinite number of times. The probability that the fourth head appears at the tenth toss is

A fair coin is tossed repeatedly till both head and tail appear at least once. The average number of tosses required is

Parcels from sender S to receiver R pass sequentially through two post-offices. Each post-office has a probability   of losing an incoming parcel, independently of all other parcels. Given that a parcel is lost, the probability that it was lost by the second post-office is_________.

Let U and V be two independent zero mean Gaussian random variables of variances  and  respectively. The probability  is        

Consider two identically distributed zero-mean random variables U and V. Let the cumulative distribution functions of U and 2V be F(x) and G(x) respectively. Then, for all values of x

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        

A fair coin is tossed till a head appears for the first time. The probability that the number of required tosses is odd, is

A fair dice is tossed two times. The probability that the second toss results in a value that is higher than the first toss is

A fair coin is tossed independently four times. The probability of the event “the number of time heads shown up is more than the number of times tails shown up” is

A fair coin is tossed 10 times. What is the probability that ONLY the first two tosses will yield heads?

Consider two independent random variables X and Y with identical distributions. The variables X and Y take values 0, 1 and 2 with probabilities ,  and  respectively. What is the conditional probability?

A discrete random variable X takes values from 1 to 5 with probabilities as shown in the table. A student calculates the mean of X as 3.5 and her teacher calculates the variance of X as 1.5. Which of the following statements is true?        

The probability density function (PDF) of a random variable X is as shown below.

The corresponding cumulative distribution function (CDF) has the form

 is the probability density function for the real random variable X, over the entire x axis. M and N are both positive real numbers. The equation relating M and N is        

If E denotes expectation, the variance of a random variable X is given by

An examination consists of two papers, Paper 1 and Paper 2. The probability of failing in Paper 1 is 0.3 and that in Paper 2 is 0.2. Given that a student has failed in Paper 2, the probability of failing in Paper 1 is 0.6. The probability of a student failing in both the papers is

A probability density function is of the form , . The value of K is

Three companies, X Y and Z supply computers to a university. The percentage of computers supplied by them and the probability of those being defective are tabulated below.

Given that a computer is defective, the probability that it was supplied by Y is:

A fair dice is rolled twice. The probability that an odd number will follow an even number is

The distribution function  of a random variable X is shown in Figure. The probability that  is