Probability and Statistics
Exam Duration: 45 Mins Total Questions : 30
There are two containers, with one containing 4 red and 3 green balls and other containing 3 blue and 4 green balls. One ball is drawn at random from each container. The probability that one of the the balls is red and the other is blue will be
- (a)
17
- (b)
\(\frac { 9 }{ 49 } \)
- (c)
\(\frac { 12 }{ 49 } \)
- (d)
\(\frac { 3 }{ 7 } \)
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)
\(\frac { 1 }{ 2 } \)
Three values of x and y are to be fitted in a straight line in the form y=1+bx by the method of least squares. Give \(\sum { x=6, } \sum { y=21, } \sum { { x }^{ 2 } } =14\quad and\quad \sum { xy=46 } \)the value of a and b are respectively
- (a)
2 and 3
- (b)
1 and 2
- (c)
2 and 1
- (d)
3 and 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)
\(\frac { 5 }{ 8 } \)
A fair coin is tossed three times in succession. If the first toss produces a head, then the probability of getting exactly two heads in three tosses is
- (a)
\(\frac { 1 }{ 8 } \)
- (b)
\(\frac { 1 }{ 2 } \)
- (c)
\(\frac { 3 }{ 8 } \)
- (d)
\(\frac { 3 }{ 4 } \)
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)
\(\frac { 3 }{ 4 } \)
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
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
For each element in a set of size 2n, an unbiased coin is tossed. The 2n coins are tossed independently. An element is chosen, if the corresponding coin toss were head. The probability that exactly n elements are chosen, is
- (a)
\(\frac { (\begin{matrix} 2n \\ n \end{matrix}) }{ { 4 }^{ n } } \)
- (b)
\(\frac { (\begin{matrix} 2n \\ n \end{matrix}) }{ { 2 }^{ n } } \)
- (c)
\(\frac { 1 }{ { (\begin{matrix} 2n \\ n \end{matrix}) } } \)
- (d)
\(\frac { 1 }{ { 2 } } \)
Two n bit binary string S1 and S2 are chosen randomly with uniform probability. The probability that the Hamming distance between these strings (the number of bit positions where are two strings differ) is equal to d is
- (a)
\(\frac { ^{ n }{ C }_{ d } }{ { 2 }^{ n } } \)
- (b)
\(\frac { ^{ n }{ C }_{ d } }{ { 2 }^{ d } } \)
- (c)
\(\frac { d }{ { 2 }^{ n } } \)
- (d)
\(\frac { 1 }{ { 2 }^{ d } } \)
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 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)
\(\frac { 5 }{ 16 } \)
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
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
A class of first year B.tech students is composed of four batches A, B, C and D, each consisting of 30 students. It is found that the sessional marks of students in Engineering drawing in batch C have a mean of 6.6 and standard deviation of 2.3. The mean and standard deviation of the marks for the entire class are 5.5 and 4.2 respectively. It is decided by the course instructer to normalize the mark o9f the students of all batches to have the same mean and standard deviation as that of the entire class. Due to this, the mark of a student in batch C are changed from 8.5 to
- (a)
6.0
- (b)
7.0
- (c)
8.0
- (d)
9.0
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
Let X be a random variable following normal distribution with mean +1 and variance 4. Let, Y be another normal variable with -1 and variance unknown. If P(X\(\le \)-1)=P(Y\(\ge \)2), the standard deviation of Y is
- (a)
3
- (b)
2
- (c)
\(\sqrt { 2 } \)
- (d)
1
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