GATE Engineering Mathematics - Probability and Statistics
Exam Duration: 45 Mins Total Questions : 30
Two coins are simultaneously tossed. The probability of two heads simultaneously appearing is
- (a)
\(\frac { 1 }{ 8 } \)
- (b)
\(\frac { 1 }{ 6 } \)
- (c)
\(\frac { 1 }{ 4 } \)
- (d)
\(\frac { 1 }{ 2 } \)
The standard deviation of spot speed of vehicles in, a highway is 8.8km/h and the mean speed of the vehicles is 33km/h the coefficient of variation in speed is
- (a)
0.1517
- (b)
0.1867
- (c)
0.2666
- (d)
0.3646
A hydraulic structure has four gates which operate independently. The probability of failure of each gate is 0.2. Given that gate 1 has failed, the probability that both gates 2 and 3 will fail is
- (a)
0.240
- (b)
0.200
- (c)
0.040
- (d)
0.008
If P and Q are two random events, then which of the following is true?
- (a)
Independence of P and Q implies that probability (P\(\cap \)Q)=0
- (b)
Probability (P\(\cup \)Q)\(\ge \)Probability (P)+Probability (Q)
- (c)
If P and Q are mutually exclusive, then they must be independent
- (d)
Probability (P\(\cap \)Q)\(\le \)Probability (P)
A box contains 4 white balls and 3 red balls. In succession, two balls randomly selected and removed from the box. Given that the first removed ball is white, the probability that the second removed ball is red, is
- (a)
\(\frac { 1 }{ 3 } \)
- (b)
\(\frac { 3 }{ 7 } \)
- (c)
\(\frac { 1 }{ 2 } \)
- (d)
\(\frac { 4 }{ 7 } \)
Assume for simplicity that N people, all born in April (a month of 30 days) are collected in a room. Consider, the event of atleast two people in the room being born on the same date of the month, even in different years, e.g., 1980 and 1985. What is the smallest N. So that the probability of this event exceeds 0.5?
- (a)
20
- (b)
7
- (c)
15
- (d)
16
A loaded dice has following probability distribution of occurrences
Dice value | 1 | 2 | 3 | 4 | 5 | 6 |
Probability | \(\frac { 1 }{ 4 } \) | \(\frac { 1 }{ 8 } \) | \(\frac { 1 }{ 8 } \) | \(\frac { 1 }{ 8 } \) | \(\frac { 1 }{ 8 } \) | \(\frac { 1 }{ 4 } \) |
If three identical dice as the above are thrown, the probability of occurrence of values 1, 5 and 6 on the three dice is
- (a)
same as that of occurrence 3, 4, 5
- (b)
same as that of occurrence 1, 2, 5
- (c)
\(\frac { 1 }{ 128 } \)
- (d)
\(\frac { 5 }{ 8 } \)
If a fair coin is tossed four times. What is the probability that two heads and two tails will results?
- (a)
\(\frac { 3 }{ 8 } \)
- (b)
\(\frac { 1 }{ 2 } \)
- (c)
\(\frac { 5 }{ 8 } \)
- (d)
\(\frac { 3 }{ 4 } \)
Let P(E) denotes the probability of the event E. Given, P(A)=1, P(B)=\(\frac { 1 }{ 2 } \), the values of \(P(\frac { A }{ B } )\) and \(P(\frac { B }{ A } )\) respectively are
- (a)
\(\frac { 1 }{ 4 } ,\frac { 1 }{ 2 } \)
- (b)
\(\frac { 1 }{ 2 } ,\frac { 1 }{ 4 } \)
- (c)
\(\frac { 1 }{ 2 } ,1\)
- (d)
\(1,\frac { 1 }{ 2 } \)
n couples are invited to a party with the condition that every husband should be accompanied by his wife. However a wife need not be accompanied by her husband. The number of different gathering possible at the party is
- (a)
\((\frac { 2n }{ n } )*{ 2 }^{ n }\quad \)
- (b)
\({ 3 }^{ n }\)
- (c)
\(\frac { (2n)! }{ { 2 }^{ n } } \)
- (d)
\((\frac { 2n }{ n } )\)
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 desk. What is the probability that the two cards are selected with the number of the first card being one higher than the number on the second card?
- (a)
\(\frac { 1 }{ 5 } \)
- (b)
\(\frac { 4 }{ 25 } \)
- (c)
\(\frac { 1 }{ 4 } \)
- (d)
\(\frac { 2 }{ 5 } \)
Consider a company that assembles computers. The probability of a faulty assembly of any computer is P. The company therefore, subject 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?
- (a)
pq+(1-p)(1-q)
- (b)
(1-q)p
- (c)
(1-p)q
- (d)
pq
An unbalanced dice (with 6 faces, number 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 option is closest to the probability that the face value exceeds 3?
- (a)
0.4533
- (b)
0.468
- (c)
0.485
- (d)
0.492
Aishwarya studies either computer science or mathematics every day. If she studies computer science on a day, then the probability that she studies mathematics the next day is 0.6. If she studies mathematics on a day, then the probability that she 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?
- (a)
0.24
- (b)
0.36
- (c)
0.4
- (d)
0.6
A program consists of two modulus executed sequentially. Let f1(t) and f2(t) respectively denote the probability density functions of time taken to execute the two modulus. The probability density function of the overall line taken to execute the program is given by
- (a)
f1(t)+f2(t)
- (b)
\(\int _{ 0 }^{ t }{ { { f }_{ 1 }(x).{ f }_{ 2 }(x)dx } } \)
- (c)
\(\int _{ 0 }^{ t }{ { { f }_{ 1 }(x).{ f }_{ 2 }(t-x)dx } } \)
- (d)
max {f1(t),f2(t)}
A fair coin is tossed 10 times. What is the probability that only the first two tosses will yield heads?
- (a)
\({ (\frac { 1 }{ 2 } ) }^{ 2 }\)
- (b)
\(^{ 10 }{ C }_{ 2 }{ (\frac { 1 }{ 2 } ) }^{ 2 }\)
- (c)
\({ (\frac { 1 }{ 2 } ) }^{ 10 }\)
- (d)
\(^{ 10 }{ C }_{ 2 }{ (\frac { 1 }{ 2 } ) }^{ 10 }\)
A probability density function is of the form P(x)=\(k{ e }^{ -\alpha \left| x \right| },x\in (-\infty ,\infty )\). The value of k is
- (a)
0.5
- (b)
1
- (c)
0.5\(\alpha \)
- (d)
\(\alpha \)
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)
\(\frac { 1 }{ 16 } \)
- (b)
\(\frac { 1 }{ 8 } \)
- (c)
\(\frac { 1 }{ 4 } \)
- (d)
\(\frac { 5 }{ 16 } \)
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)
\(\frac { 2 }{ 36 } \)
- (b)
\(\frac { 2 }{ 6 } \)
- (c)
\(\frac { 5 }{ 12 } \)
- (d)
\(\frac { 1 }{ 2 } \)
An examination consists of two papers, Paper 1, 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 paper is
- (a)
0.5
- (b)
0.18
- (c)
0.12
- (d)
0.06
The mean of a set of number is \(\overline { x } \). If each number is increased by \(\lambda \), then variance of the new set is
- (a)
\(\overline { x } \)
- (b)
\(\overline { x } +\lambda \)
- (c)
\(\lambda \overline { x } \)
- (d)
None of these
If \(\mu \) is mean of distribution, then \(\Sigma { f }_{ i }(Y_{ i }-\mu )\) is equal to
- (a)
MD
- (b)
Standard deviation
- (c)
0
- (d)
None of these
bxyxbyx is equal to
- (a)
\(\rho \)(X,Y)
- (b)
Cov(X,Y)
- (c)
{\(\rho \)(x,Y)}2
- (d)
None of these
If two lines of regression are at right angles, then \(\rho \) (X,Y) is equal to
- (a)
1
- (b)
-1
- (c)
1 or -1
- (d)
0
In a frequency distribution, mid value of a class is 15 and class interval is 4. The lower limit of the class is
- (a)
14
- (b)
13
- (c)
12
- (d)
10
A has one share in a lottery in which there is 1 prize and 2 blanks.B has three shares in a lottery in which there are 3 prizes and 6 blanks compare the probability of A's success of that of B's success is
- (a)
7:16
- (b)
16:7
- (c)
6:14
- (d)
14:6
What will be the mean and standard deviation for the following table given the age distribution of 542 members
Age in years | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | 80-90 |
Number of members | 3 | 61 | 132 | 153 | 140 | 51 | 2 |
- (a)
54.72 and 11.9
- (b)
50 and 43.1
- (c)
43.5 and 31.8
- (d)
19 and 10
Examination paper has 150 multiple choice questions of one mark each question having four choices. Each incorrect answer fetches -0.25 mark. Suppose 1000 students choose all their answer randomly with uniform probability. the sum total of the expected marks obtained by all these students
- (a)
0
- (b)
2550
- (c)
7525
- (d)
9375
The odds against a husband who is 45 yr old, living till he is 70 are 7:5 and the odds against his wife who is 36, living till she is 61 are 5:3. The probability that atleast one of them will be alive, 25 yr hence, is
- (a)
\(\frac { 61 }{ 96 } \)
- (b)
\(\frac { 5 }{ 36 } \)
- (c)
\(\frac { 13 }{ 64 } \)
- (d)
None of these
Ten students got the following percentage of marks in Economics and Statistics
Roll number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
Marks in Economics | 78 | 36 | 98 | 25 | 75 | 82 | 90 | 62 | 65 | 39 |
Marks in statistics | 84 | 51 | 91 | 60 | 68 | 62 | 86 | 58 | 53 | 47 |
The coefficient of correlation is
- (a)
0.78
- (b)
7.8
- (c)
2.3
- (d)
78