GATE CS Probability: Random Variables and Distributions questions

Practise Probability: Random Variables and Distributions questions for GATE CS. Explore official PYQs and original topic-wise quizzes, check answers and view available solutions. Free, no sign-in.

Define the sample space and events first. Distinguish independent events from conditional probabilities, and check that the final probability lies between zero and one.

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SubjectTopicYearType

Suppose we uniformly and randomly select a permutation from the

20 !

permutations of

1, 2, 3\ldots ,20.

What is the probability that

2

appears at an earlier position than any other even number in the selected permutation?

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 was a head. The probability that exactly

n

elements are chosen is

We are given a set

X = \{X_1,\ldots,X_n\}

X_i=2^i

S\subseteq X

is drawn by selecting each

X_i

independently with probability

P_i = \frac{1}{2}

. The expected value of the smallest number in sample

S

A random bit string of length n is constructed by tossing a fair coin n times and setting a bit to 0 or 1 depending on outcomes head and tail, respectively. The probability that two such randomly generated strings are not identical is:

f(x)

be the continuous probability density function of a random variable

x

, the probability that

a < x \leq b

A point is randomly selected with uniform probability in the

X-Y

plane within the rectangle with corners at

(0,0), (1,0), (1,2)

(0,2).

p

is the length of the position vector of the point, the expected value of

p^{2}

n

bit binary strings,

S_1

S_2

are chosen randomly with uniform probability. The probability that the Hamming distance between these strings (the number of bit positions where the two strings differ) is equal to

d

An examination paper has

150

multiple choice questions of one mark each, with each question having four choices. Each incorrect answer fetches

-0.25

1000

students choose all their answers randomly with uniform probability. The sum total of the expected marks obtained by all these students is

If a fair coin is tossed four times. What is the probability that two heads and two tails will result?

A program consists of two modules executed sequentially. Let

f_1(t)

f_2(t)

respectively denote the probability density functions of time taken to execute the two modules. The probability density function of the overall time taken to execute the program is given by

Four fair coins are tossed simultaneously. The probability that at least one head and one tail turn up is

Consider two events

E_1

E_2

such that probability of

E_1

P_r[E_1]=\frac{1}{2}

, probability of

E_2

P_r[E_{2}]=\frac{1}{3}

, and probability of

E_1

E_2

P_r[E_1 \: and \: E_2] = \frac{1}{5}

. Which of the following statements is/are true?

Suppose that the expectation of a random variable

X

5

. Which of the following statements is true?

Two friends agree to meet at a park with the following conditions. Each will reach the park between 4:00 pm and 5:00 pm and will see if the other has already arrived. If not, they will wait for 10 minutes or the end of the hour whichever is earlier and leave. What is the probability that the two will not meet?

A die is rolled three times. The probability that exactly one odd number turns up among the three outcomes is

The probability that it will rain today is

0.5

. The probability that it will rain tomorrow is

0.6

. The probability that it will rain either today or tomorrow is

0.7

. What is the probability that it will rain today and tomorrow?

The probability that top and bottom cards of a randomly shuffled deck are both aces is

Two dice are thrown simultaneously. The probability that at least one of them will have

6

10

white balls and

15

black balls. Two balls are drawn in succession. The probability that one of them is black and the other is white is:

The probability that a number selected at random between

100

999

(both inclusive) will not contain the digit

7