Engineering Mathematics - 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)
\(\frac { 1 }{ 2 } \)
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
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 the difference between the exceptation of the square of random variable and the square of the expectation of the random variable is denoted by R,
- (a)
R=0
- (b)
R<0
- (c)
R\(\ge \)0
- (d)
R>0
Let, f(x) be the continuous probability density function of a random variable X. The probability that a<X\(\le \)b is
- (a)
f(b-a)
- (b)
f(b)-f(a)
- (c)
\(\int _{ a }^{ b }{ f(x)dx } \)
- (d)
\(\int _{ a }^{ b }{ xf(x)dx } \)
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)
\((\frac { 2n }{ n } )\)
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
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 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 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 mode of a data is 18 and mean is 24, then median
- (a)
18
- (b)
24
- (c)
22
- (d)
21
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 person on a trip has a choice between private car and public transport.The probability of using a private car is 0.45. While using public transport, further choice available are bus and metro, out of which the probability of commuting by a bus is 0.55. In such a situation, the probability ( rounded up to two decimals ) of using a car, bus and metro, respectively would be
- (a)
0.45, 0.30 and 0.25
- (b)
0.45, 0.25 and 0.30
- (c)
0.45, 0.55 and 0
- (d)
0.45, 0.35, 0.20
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
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
A point is randomly selected with uniform probability in the X-Y plane within the rectangular with corners at (0,0), (1,0), (1,2) and (0,2). If P is the length of the position vector of the point, the expected value of P2 is
- (a)
\(\frac { 2 }{ 3 } \)
- (b)
1
- (c)
\(\frac { 4 }{ 3 } \)
- (d)
\(\frac { 4 }{ 3 } \)
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
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