Probability-and-Statistics

Question 1
The lifetime of a component of a certain type is a random variable whose probability density function is exponentially distributed with parameter 2. For a randomly picked component of this type, the probability that its lifetime exceeds the expected lifetime (rounded to 2 decimal places) is _______.
A
0.37
Engineering-MathematicsProbability-and-statisticsGATE 2021 CS-Set-1Video-Explanation
Question 1 Explanation: 
Question 2
Consider the two statements.
           S1: There exist random variables X and Y such that
                           EX-E(X)Y-E(Y)2>Var[X] Var[Y]
           S2: For all random variables X and Y,
                            CovX,Y=E|X-E[X]| |Y-E[Y]| 
Which one of the following choices is correct?
A
S1is false, but S2is true.
B
Both S1and S2are true.
C
S1is true, but S2is false.
D
Both S1and S2 are false.
Engineering-MathematicsProbability-and-statisticsGATE 2021 CS-Set-1Video-Explanation
Question 2 Explanation: 
Variance(X) = Var[X]= E((X-E(X))^2)
For a dataset with single values, we have variance 0. EX-E(X)Y-E(Y)2>Var[X] Var[Y]
This leads to inequance of 0>0 which is incorrect.

Its not |x-E(x)|. Thus S2 is also incorrect.
Question 3
A sender (S) transmits a signal, which can be one of the two kinds: H and L with probabilities 0.1 and 0.9 respectively, to a receiver (R). In the graph below, the weight of edge (u, v) is the probability of receiving v when u is transmitted, where u, v ∈ {H, L}. For example, the probability that the received signal is L given the transmitted signal was H, is 0.7.

If the received signal is H, the probability that the transmitted signal was H (rounded to 2 decimal places) is _______
A
0.04
Engineering-MathematicsProbability-and-statisticsGATE 2021 CS-Set-1Video-Explanation
Question 3 Explanation: 

Bayes theorem:
Probability of event A happening given that event B has already happened is
P(A/B) = P(B/A)*P(A)  / P(B)

Here, it is asked that P( H transmitted / H received).

S can send signal  to H with 0.1 probability, S can send signal to L with 0.9 probability.
The complete diagram can be



Probability that H Transmitted (H_t) given that H received (H_r)is

P( H_t  / H_r) = P( H_r/ H_t) * P(H_t)  / P(H_r)

P(H-r) = probability that H received  = P( H received from H)+ P(H received from L)
It can be observed from the graph that H can receive in two ways (S to H to H) and (S to L to H)
The P(H_r) = 0.1*0.3 + 0.9*0.8= 0.03+0.72 = 0.75

P(H_received given that H_transmitted) =0.3
P(H transmitted ) = 0.1  i.e.

P( H_t  / H_r) = P( H_r/ H_t) * P(H_t)  / P(H_r)
                        = 0.3*0.1 / 0.75 = 0.04

 

Question 4

(a) Two friends agree to meet at a park with the following conditions. Each will reach the park between 4:00 p.m. and 5:00 p.m. 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?

(b) Given a regular expression for the set of binary strings where every 0 is immediately followed by exactly k 1's and preceded by atleast k 1's (k is a fixed integer).

A
Theory Explanation.
Engineering-MathematicsProbability-and-statisticsGATE 1998
Question 5
For a given biased coin, the probability that the outcome of a toss is a head is 0.4. This coin is tossed 1,000 times. Let X denote the random variable whose value is the number of times that head appeared in these 1,000 tosses. The standard deviation of X (rounded to 2 decimal places) is _______.
A
15.49
Engineering-MathematicsProbability-and-statisticsGATE 2021 CS-Set-2Video-Explanation
Question 5 Explanation: 
Question 6
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 QuesB 0.8 0.5 10 20
Assuming that the student always wants to maximize her expected marks in the examination, in which order should she attempt the questions and what is the expected marks for that order (assume that the questions are independent)?
A
First QuesB and then QuesA. Expected marks 22.
B
First QuesA and then QuesB. Expected marks 16.
C
First QuesA and then QuesB. Expected marks 14.
D
First QuesB and then QuesA. Expected marks 14.
Engineering-MathematicsProbability-and-statisticsGATE 2021 CS-Set-2Video-Explanation
Question 6 Explanation: 

There are two ways to get marks.
1. Answering first question correctly, and second one wrongly
2. Answering first question and second question correctly.

There are two ways to answer the question paper. First A or First B.
In total we get 4 ways of answering the paper to get marks. 

Note: Answering each question is independent, thus P(x intersection y) = P(x)*P(y)|
P(Answering A correctly) =0.8, P( Answering A wrongly) = 1-0.8=0.2
P(Answering B correctly) =0.5, P( Answering B wrongly) = 1-0.5=0.5



 

Probability of Getting marks 

Marks 

First A correctly, B wrongly

P(A correctly)*P(B wrongly)
= 0.8*0.5 = 0.4

10+0 = 10

First A correctly B correctly

P(A correctly)*P(B Correctly)
= 0.8*0.5 = 0.4

10+20 = 30

First B correctly, A wrongly

P(B correctly)*P(A wrongly)
= 0.5*0.2 = 0.1

20+0=20

First B correctly then A correctly

P(B correctly)*P(A Correctly)
=0.5*0.8 = 0.4

20+10=30

 

Expectation formula
                                   

                                         E(X)=∑X*P(X)

 

   Expectation for the order:A followed by B = 0.4*10 +0.4*30 = 16
Expectation for the order B followed by A = 0.1*20 + 0.4*30 = 14

 

As the target is to get maximum marks, the order A followed by B is the correct option

Question 7
A bag has r red balls and b 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?
A
B
C
D
E
Engineering-MathematicsProbability-and-statisticsGATE 2021 CS-Set-2Video-Explanation
Question 7 Explanation: 

  • There are 8 ways to get ‘red’ in the fourth attempt of drawing 

    i.e., _ _ _ R these three gaps can be filled with B/R in 222=8 ways

    The probability for case (1),


Question 8
The number of arrangements of six identical balls in three identical bins is______.
A
7
Engineering-MathematicsProbability-and-statisticsGATE 2022Video-Explanation
Question 8 Explanation: 
Given distribution of n identical items into r identical boxes.
As no other condition is given,
We need to consider that distinct distribution will be based on the count of balls.
6 identical balls into 3 identical bins.
That can be done in the combination of
[6,0,0], [5,1,0], [ 4,2,0], [3,3,0], [4,1,1],[3,2,1],[2,2,2]
I.e. in 7 ways
Question 9
Consider a random experiment where two fair coins are tossed. Let A be the event that denotes HEAD on both the throws, B be the event that denotes HEAD on the first throw, and C be the event that denotes HEAD on the second throw. Which of the following statements is/are TRUE?
A
A and B are independent.
B
A and C are independent.
C
B and C are independent.
D
Prob(B|C) = Prob(B)
Engineering-MathematicsProbability-and-statisticsGATE 2023Video-Explanation
Question 10
Let U = {1, 2,...,n}, where n is a large positive integer greater than 1000. Let k be a positive integer less than n. Let A, B be subsets of U with |A| = |B| = k and A ⋂ B = ∅. We say that a permutation of U separates A from B if one of the following is true.
- All members of A appear in the permutation before any of the members of B.
- All members of B appear in the permutation before any of the members of A.
How many permutations of U separate A from B?
A
A
B
B
C
C
D
D
Engineering-MathematicsProbability-and-statisticsGATE 2023Video-Explanation
Question 11

Consider a finite sequence of random values X = [x1, x2, …, 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
Index position of mode of X in X is the same as the index position of mode of Y in Y.
B
Index position of median of X in X is the same as the index position of median of Y in Y.
C
μy = aμx + b
D
σy = aσx + b
Engineering-MathematicsProbability-and-StatisticsGATE 2011
Question 11 Explanation: 
σy is standard deviation then
(σy)2 is variance so,
yi = a * xi + b
(σy)2 = a2 (σx)2
⇒ σy = a σx
Hence option (D) is incorrect.
Question 12
A
a
B
b
C
c
D
d
Engineering-MathematicsProbability-and-statisticsCMI 2019
Question 12 Explanation: 
For i ϵ {1,...,n}, let Pi be the set of all permutations x1;x2,..., xn for which x1 > x2 >... > xi.
Question 13
Suppose you alternate between throwing a normal six-sided fair die and tossing a fair coin. You start by throwing the die. What is the probability that you will see a 5 on the die before you see tails on the coin?
A
1/12
B
1/6
C
2/9
D
2/7
Engineering-MathematicsProbability-and-statisticsCMI 2019
Question 13 Explanation: 
Question 14
Alice and Bob have one coin each with probability of Heads p and q, respectively. In each round, both Alice and Bob independently toss their coin once, and the game stops if one of them gets a Heads and the other gets a Tails. If they both get either Heads or both get Tails in any round, the game continues. Let R be the expected number of rounds by which the game stops. Which of the following is TRUE?
A
R=1/(p+q-2pq)
B
R=1/(p+q-p2q
C
R is independent of p and q
D
R=1/(1+2pq-p-q)
E
None of the above
Engineering-MathematicsProbability-and-statisticsTIFR PHD CS & SS 2020
Question 15
Consider two independent random variables (U1, U2) both are uniformly distributed between [0, 1]. The conditional expectation
E[(U1 + U2)| max(U1, U2) ≥ 0.5]
equals
A
7/6
B
8/7
C
6/7
D
1.1
E
None of the above
Engineering-MathematicsProbability-and-statisticsTIFR PHD CS & SS 2020
Question 16
Suppose that X is a real valued random variable and E[exp X] = 2. Then, which of the following must be TRUE?
Hint: (exp(x) + exp(y))/2 ≥ exp((x + y)/2).
A
E[X] < ln 2
B
E[X] > ln 2
C
E[X] ≥ ln 2
D
E[X] ≤ ln 2
E
None of the above
Engineering-MathematicsProbability-and-statisticsTIFR PHD CS & SS 2020
Question 17
Consider a unit disc D. Let a point x be chosen uniformly on D and let the random distance to x from the center of D be R. Which of the following is TRUE?
A
R2 is uniformly distributed in [0, 1]
B
πR2 is uniformly distributed in [0, 1]
C
π R2 is uniformly distributed in [0, 1]
D
2πR2 is uniformly distributed in [0, 1]
E
None of the above
Engineering-MathematicsProbability-and-statisticsTIFR PHD CS & SS 2020
Question 19
Suppose that Dice 1 has five faces numbered 1 to 5, each of which is equally likely to occur once the dice is rolled. Dice 2 similarly has eight equally likely faces numbered 1 to 8. Suppose that the two dice are rolled, and the sum is equal to 8. Conditioned on this, what is the chance that the Dice 1 rolled a number less than or equal to 2?
A
1/4
B
1/3
C
1/2
D
2/7
E
2/5
Engineering-MathematicsProbability-and-statisticsTIFR PHD CS & SS 2020
Question 20
Balls are drawn one after the other uniformly at random without replacement from a set of eight balls numbered 1, 2, . . . , 8 until all balls drawn. What is the expected number of balls whose value match their ordinality (i.e., their position in the order in which balls were drawn)?
Hint: what is the probability that the i-th ball is drawn at the i-th draw? Now can you use linearity of expectation to solve the problem?
A
1
B
1.5
C
2
D
2.5
E
None of the above
Engineering-MathematicsProbability-and-statisticsTIFR PHD CS & SS 2020
Question 21
Suppose we toss m = 3 labelled balls into n = 3 numbered bins. Let A be the event that the first bin is empty while B be the event that the second bin is empty. P(A) and P(B) denote their respective probabilities. Which of the following is true?
A
P(A) > P(B)
B
P(A) = 1/27
C
P(A) > P(A|B)
D
P(A) < P(A|B)
E
None of the above
Engineering-MathematicsProbability-and-statisticsTIFR PHD CS & SS 2020
Question 22
A lottery chooses four random winners. What is the probability that at least three of them are born on the same day of the week? Assume that the pool of candidates is so large that each winner is equally likely to be born on any of the seven days of the week independent of the other winners
A
17 / 2401
B
48 / 2401
C
105 / 2401
D
175 / 2401
E
294 / 2401
Engineering-MathematicsProbability-and-statisticsTIFR PHD CS & SS 2020
Question 24
Two balls are drawn uniformly at random without replacement from a set of five balls numbered 1, 2, 3, 4, 5. What is the expected value of the larger number on the balls drawn?
A
2.5
B
3
C
3.5
D
4
E
None of the above
Engineering-MathematicsProbability-and-statisticsTIFR PHD CS & SS 2020
Question 25
A drunken man walks on a straight lane. At every integer time (in seconds) he moves a distance of 1 unit randomly, either forwards or backwards. What is the expectation of the square of the distance after 100 seconds from the initial position?
Hint: The position at time 100 is a sum of independent and identically distributed random variables.
A
100
B
C
40
D
200
E
Engineering-MathematicsProbability-and-statisticsTIFR PHD 2022
Question 26
Let a bag contain ten balls numbered 1, 2, . . . , 10. Let three balls be drawn at random in sequence without replacement, and the number on the ball drawn on the ith choice be ni∈ {1, 2, . . . , 10}. What is the probability that n_1 < n_2 < n_3?
A
1/3
B
1/12
C
1/4
D
1/6
E
None of the above
Engineering-MathematicsProbability-and-statisticsTIFR PHD 2022
Question 27
Suppose you throw a dart and it lands uniformly at random on a target which is a disk of unit radius. What is the probability density function f(x) of the distance of the dart from the center of the disk?
A
B
C
D
E
None of the above.
Engineering-MathematicsProbability-and-statisticsTIFR PHD 2022
Question 28
Two players A and B of equal skill are playing a match. The first one to win 4 rounds wins the match. Both players are equally likely to win each round independent of the outcomes of the other rounds. After 3 rounds, A has won 2 rounds and B has won 1 round. Conditioned on this, what is the conditional probability that A wins the match?
A
5/8
B
2/3
C
11/16
D
5/7
E
None of the above
Engineering-MathematicsProbability-and-statisticsTIFR PHD 2022
Question 29
Consider a coin flip game between Amar, Akbar and Anthony. A fair coin (so that heads and tails each have probability 0.5) is independently flipped five times. Amar wins if at least three consecutive draws of heads are observed in the five coin tosses. Akbar wins if at least three consecutive draws of tails are observed in the five coin tosses. Anthony wins if the other two do not win. What is the probability of Anthony winning?
A
9/16
B
1/3
C
1/2
D
5/8
E
7/12
Engineering-MathematicsProbability-and-statisticsTIFR PHD 2022
Question 30
Suppose that a random variable X can take 5 values {1, 2, 3, 4, 5} with probabilities that depend upon n ≥ 0 and are given by

for k = 1, 2, 3, 4, 5. What can one say about the expectation E[X] as n → ∞?
A
It increases to infinity as n → ∞
B
It equals 3 for all values of n ≥ 0
C
It converges to 1 as n → ∞
D
It converges to 5 as n → ∞
E
It converges to 0 as n → ∞
Engineering-MathematicsProbability-and-statisticsTIFR PHD 2022
Question 31
Let F be the set of all functions mapping {1, . . . , n} to {1, . . . ,m}. Let f be a function that is chosen uniformly at random from F. Let x, y be distinct elements from the set {1, . . . , n}. Let p denote the probability that f(x) = f(y). Then,
A
p = 0
B
C
D
E
Engineering-MathematicsProbability-and-statisticsTIFR PHD 2022
There are 31 questions to complete.

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