NAT2 marks
Let be a random variable which takes values in the set . Further, and .
The expected value of , denoted by , is equal to ___________. (*rounded off to two decimal places*)
Numerical answer
NAT2 marks
Suppose an unbiased coin is tossed 6 times. Each coin toss is independent of all previous coin tosses. Let be the event that among the second, fourth, and sixth coin tosses, there are at least two heads. Let be the event that among the first, second, third, and fifth coin tosses, there are equal number of heads and tails.
The conditional probability is equal to ____________. (*rounded off to one decimal place*)
Numerical answer
NAT2 marks
Suppose a 5-bit message is transmitted from a source to a destination through a noisy channel. The probability that a bit of the message gets flipped during transmission is 0.01. Flipping of each bit is independent of one another. The probability that the message is delivered error-free to the destination is ______. (rounded off to three decimal places)
Numerical answer
NAT2 marks
Consider a probability distribution given by the density function .
The probability that lies between 2 and 3, i.e., is __________. (rounded off to three decimal places)
Numerical answer
NAT2 marks
A quadratic polynomial over complex numbers is said to be square invariant if . Suppose from the set of all square invariant quadratic polynomials we choose one at random.
The probability that the roots of the chosen polynomial are equal is __________. (rounded off to one decimal place)
Numerical answer
NAT2 marks
The unit interval is divided at a point chosen uniformly distributed over in into two disjoint subintervals.
The expected length of the subinterval that contains 0.4 is ___________. (rounded off to two decimal places)
Numerical answer
NAT2 marks
A bag contains 10 red balls and 15 blue balls. Two balls are drawn randomly without replacement. Given that the first ball drawn is red, the probability (rounded off to 3 decimal places) that both balls drawn are red is __________
Numerical answer
MCQ2 marks
Let and be random variables, not necessarily independent, that take real values in the interval . Let and let the mean values of be , respectively. Which one of the following statements is TRUE?
Single correct
MSQ2 marks
Consider a random experiment where two fair coins are tossed. Let be the event that denotes HEAD on both the throws, be the event that denotes HEAD on the first throw, and be the event that denotes HEAD on the second throw.
Which of the following statements is/are TRUE?
One or more correct
Answer key coming soon
MCQ2 marks
Consider the two statements.
: There exist random variables and such that
: For all random variables and ,
Which one of the following choices is correct?
Single correct
Answer key coming soon
NAT2 marks
A sender (S) transmits a signal, which can be one of the two kinds: and with probabilities 0.1 and 0.9 respectively, to a receiver (R).
In the graph below, the weight of edge is the probability of receiving when is transmitted, where . For example, the probability that the received signal is given the transmitted signal was , is 0.7.
If the received signal is , the probability that the transmitted signal was (rounded to 2 decimal places) is __________.
Numerical answer
Answer key coming soon
MCQ2 marks
In an examination, a student can choose the order in which two questions (
QuesA and QuesB) must be attempted.
- If the first question is answered wrong, the student gets zero marks.
- If the first question is answered correctly and the second question is not answered correctly, the student gets the marks only for the first question.
- If both the questions are answered correctly, the student gets the sum of the marks of the two questions.
The following table shows the probability of correctly answering a question and the marks of the question respectively.
question probability of answering correctly marks
QuesA 0.8 10
QuesB 0.5 20Single correct
Answer key coming soon
MCQ2 marks
A bag has red balls and black balls. All balls are identical except for their colours. In a trial, a ball is randomly drawn from the bag, its colour is noted and the ball is placed back into the bag along with another ball of the same colour. Note that the number of balls in the bag will increase by one, after the trial. A sequence of four such trials is conducted. Which one of the following choices gives the probability of drawing a red ball in the fourth trial?
Single correct
Answer key coming soon
NAT2 marks
For , let be a non-zero vector. Suppose that is chosen uniformly at random from . Then, the probability that is an odd number is ________.
Numerical answer
Answer key coming soon
NAT2 marks
Let be a full binary tree with 8 leaves. (A full binary tree has every level full.) Suppose two leaves and of are chosen uniformly and independently at random. The expected value of the distance between and in (i.e., the number of edges in the unique path between and ) is (rounded off to 2 decimal places) _____.
Numerical answer
NAT2 marks
Suppose is distributed uniformly in the open interval . The probability that the polynomial has only real roots is (rounded off to 1 decimal place) _____.
Numerical answer
NAT2 marks
Consider Guwahati (G) and Delhi (D) whose temperatures can be classified as high (), medium () and low (). Let denote the probability that Guwahati has high temperature. Similarly, and denotes the probability of Guwahati having medium and low temperatures respectively. Similarly, we use , and for Delhi.
The following table gives the conditional probabilities for Delhi's temperature given Guwahati's temperature.
Consider the first row in the table above. The first entry denotes that if Guwahati has high temperature () then the probability of Delhi also having a high temperature () is 0.40; i.e., . Similarly, the next two entries are and . Similarly for the other rows.
If it is known that , , and , then the probability (correct to two decimal places) that Guwahati has high temperature given that Delhi has high temperature is _____.
Numerical answer
MCQ2 marks
P and Q are considering to apply for a job. The probability that P applies for the job is , the probability that P applies for the job given that Q applies for the job is , and the probability that Q applies for the job given that P applies for the job is . Then the probability that P does not apply for the job given that Q does not apply for the job is
Single correct
Answer key coming soon
MCQ2 marks
For any discrete random variable , with probability mass function
, , , and , define the polynomial function . For a certain discrete random variable , there exists a scalar such that . The expectation of is
Single correct
Answer key coming soon
NAT2 marks
If a random variable has a Poisson distribution with mean 5, then the expectation equals ________.
Numerical answer
Answer key coming soon
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