# GATE Questions & Answers of Probability Computer Science and Information Technology

#### Probability 28 Question(s)

Two numbers are chosen independently and uniformly at random from the set {1, 2 , . . . , 13}. The probability (rounded off to 3 decimal places) that their 4-bit (unsigned) binary representations have the same most significant bit is ___________.

Suppose Y is distributed uniformly in the open interval (1,6). The probability that the polynomial $3x^2+6xY+3Y+6$ has only real roots is (rounded off to 1 decimal place) ________.

Two people, P and Q, decide to independently roll two identical dice, each with 6 faces, numbered 1 to 6. The person with the lower number wins. In case of a tie, they roll the dice repeatedly until there is no tie. Define a trial as a throw of the dice by P and Q. Assume that all 6 numbers on each dice are equi-probable and that all trials are independent. The probability (rounded to 3 decimal places) that one of them wins on the third trial is _____.

Let X be a Gaussian random variable with mean 0 and variance $\style{font-family:'Times New Roman'}{\sigma^2}$. Let Y= max(X,0) where max(a,b) is the maximum of a and b. The median of Y is ___________.

P and Q are consider to apply for a job. The probability that P applies for the job is $\frac14$, the probability that applies for the job given that Q applies for the job is $\frac12$. and the probability that Q applies for the job given that P applies for the job is $\frac13$. Then the probability that P does not apply for the job given that Q does not apply for the job is

For any discrete random variable X, with probability mass function

$P(X=j)\;=\;p_j,p_j\geq0,\;\;j\in\{0,......,N\},$ and $\sum_{j=0}^N\;p_j=1,\;$ define the polynomial function $g_X(z)=\overset N{\underset{j=0}{\sum\;}}p_jz^j$. For a certain discrete random variable Y, there exist a scalar $\style{font-family:'Times New Roman'}{\beta\in\left[0,1\right]}$ such that $g_y(z)=(1-\beta+\beta\;z)^N$. The expectation of Y is

If a random variable X has a Poisson distribution with mean 5, then the expectation E[(X+2)2] equals ______.

Consider the following experiment.

Step1. Flip a fair coin twice.
Step2. If the outcomes are (TAILS, HEADS) then output Y and stop.

Step4. If the outcomes are (TAILS, TAILS), then go to Step 1.

The probability that the output of the experiment is Y is (up to two decimal places) __________.

In a room there are only two types of people, namely Type 1 and Type 2. Type 1 people always tell the truth and Type 2 people always lie. You give a fair coin to a person in that room, without knowing which type he is from and tell him to toss it and hide the result from you till you ask for it. Upon asking, the person replies the following
“The result of the toss is head if and only if I am telling the truth.”
Which of the following options is correct?

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

Four fair six-sided dice are rolled. The probability that the sum of the results being 22 is $X}{1296}$. The value of X is _________.

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 = _____________.

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.)

The probability that a given positive integer lying between 1 and 100 (both inclusive) is NOT divisible by 2, 3 or 5 is ______ .

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

Suppose p is the number of cars per minute passing through a certain road junction between 5 PM and 6 PM, and p has a Poisson distribution with mean 3. What is the probability of observing fewer than 3 cars during any given minute in this interval ?

Consider a random variable X that takes values +1 and −1 with probability 0.5 each. The values of the cumulative distribution function F(x) at x = −1 and +1 are

Suppose a fair six-sided die is rolled once. If the value on the die is 1, 2, or 3, the die is rolled a second time. What is the probability that the sum total of values that turn up is at least 6?

If two fair coins are flipped and at least one of the outcomes is known to be a head, what is the probability that both outcomes are heads?

If the difference between the expectation of the square of a random variable $\left(E\left[{X}^{2}\right]\right)$ and the square of the expectation of the random variable ${\left(E\left[X\right]\right)}^{2}$ is denoted by R , then

Consider a finite sequence of random values X = [x1, x 2,...,xn ]. Let μx be the mean and σx be the standard deviation of X. Let another finite sequence Y of equal length be derived from this as yi = a *xi + b, where a and b are positive constants. Let μy be the mean and σy be the standard deviation of this sequence. Which one of the following statements is INCORRECT?

A deck of 5 cards (each carrying a distinct number from 1 to 5) is shuffled thoroughly. Two cards are then removed one at a time from the deck. What is the probability that the two cards are selected with the number on the first card being one higher than the number on the second card?

Consider a company that assembles computers. The probability of a faulty assembly of any computer is p. The company therefore subjects each computer to a testing process. This testing process gives the correct result for any computer with a probability of q. What is the probability of a computer being declared faulty?

What is the probability that divisor of 1099 is a multiple of 1096?

An unbalanced dice (with 6 faces, numbered from 1 to 6) is thrown. The probability that the face value is odd is 90% of the probability that the face value is even. The probability of getting any even numbered face is the same.

If the probability that the face is even given that it is greater than 3 is 0.75, which one of the following options is closest to the probability that the face value exceeds 3?

Aishwarya studies either computer science or mathematics everyday. If she studies computer science on a day, then the probability that the studies mathematics the next day is 0.6. If she studies mathematics on a day, then the probability that the studies computer science the next day is 0.4. Given that Aishwarya studies computer science on Monday, what is the probability that she studies computer science on Wednesday?