Statistics and Probability Test

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Statistics and Probability Test - 1

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Question 1
2 / -0.83

A fair dice is rolled twice. The probability that an odd number will follow an even number is

A
1/2

B
1/6

C
1/3

D
1/4

Solution

Here the sample space S = 6
Therefore P(odd number ) = 3/6 = 1/2
and P (even number ) = 3/6 = 1/2
since events are independent,
therefore, P (odd / even )
Question 2
2 / -0.83

Three companies X, Y and Z supply computers to a university. The percentage of computers supplied by them and the probability of those being defective are tabulated below
Given that a computer is defective, the probability that it was supplied by Y is

A
0. 1

B
0.2

C
0.3

D
0.4

Solution

Probability of defective computer supplied by Y =
(Case when Y produces defective)/(All cases of producing defective product)
Case when Y produces defective = (0.3)(0.02) = 0.006
All cases of producing defective product= (0.6x0.01)+(0.3x0.02)
(0.1x0.03)= 0.006+0.006+0.003=0.015
Probability = 0.006/0.015=0.4
Question 3
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An examination consists of two papers, Paper 1 and 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 papers is

A
0.5

B
0.18

C
0.12

D
0.06

Solution

Let A be the event that ‘failed in paper 1 ’.
B be the event that ‘failed in paper 2 ’.
Given P(A) = 0.3, P(B) = 0.2.
To find the probability of failing in both papers (P(F1 ∩F2)),
we use the formula for conditional probability: P(F1 | F2) = P(F1 ∩F2) / P(F2).
Here, P(F1 | F2) = 0.6, and P(F2) = 0.2.
Therefore, P(F1 ∩F2) = P(F1 | F2) ×P(F2)
= 0.6 ×0.2 = 0.12.
Thus, the correct answer isoption 3, with a probability of 0.12.
Question 4
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A fair coin is tossed 10 times. What is the probability that ONLY the first two tosses will yield heads?

A

B

C

D

Solution

Let A be the event that first toss is head
And B be the event that second toss is head.
By the given condition rest all 8 tosses should be tail
∴The probability of getting head in first two cases
Question 5
2 / -0.83

Consider the continuous random variable with probability density function
The standard deviation of the random variables is

A
1/√3

B
1/√6

C
1/3

D
1/6

Solution

T being the random variable of f(t).
Question 6
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The probability that two friends share the same birth-month is

A
1/6

B
1/12

C
1/144

D
1/24

Solution

Let A = the event that the birth month of first friend
And B= that of second friend.
∴P( A )= 1, as 1st friend can born in any month
and P(B) = 1/12, by the condition.
∴Probability of two friends share same birth-month
Question 7
2 / -0.83

A box contains 5 black balls and 3 red balls. A total of three balls are picked from the box one after another, without replacing them back. The probability of getting two black balls and one red ball is

A
3/8

B
2/15

C
15/28

D
½

Solution

Here the possible combination of picking up three balls without replacement is BBR, BRB, RBB.
(B = Black ball, R = Red balls)
9
∴Probability of getting two black balls and one red ball is (5/28)x3 = 15/28.
Question 8
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Two dice are thrown. What is the probability that is the sum of the numbers on the two dice is eight?

A
5/36

B
5/18

C
¼

D
1/3

Solution

Here sample space = 6 ×6 = 36
Here, there are five such points whose sum is 8. They are (2,6), (3,5), (4,4), (5,3), (6,2).
Question 9
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Arrivals at a telephone booth are considered to be poison, with an average time of 10 minutes between successive arrivals. The length of a phone call is distributes exponentially with mean 3 minutes. The probability that an arrival does not have to wait before service is

A
0.7

B
0.5

C
0.3

D
0.9

Solution

The correct option is a) 0.7
Given data:

λ=0.1 per minute,

Service rate, μ=0.33 per minute

ρ=0.1/0.33=0.3

Probability that an arrival does not have to wait before service or probability that a system is idle,

P₀ =1 −ρ=1 −0.3=0.7
Question 10
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A box contains 5 black and 5 red balls. Two balls are randomly picked one after another from the box, without replacement. The probability for both balls being red is

A
1/90

B
1/2

C
19/90

D
2/9

Solution

The probability of drawing two red balls without replacement
Question 11
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A lot has 10% defective items. Ten items are chosen randomly from this lot. The probability that exactly 2 of the chosen items are defective is

A
0.0036

B
0.1937

C
0.2234

D
0.3874

Solution

Let A be the event that items are defective and B be the event that items are non- defective
∴P( A )= 0.1 and P(B) = 0.9
∴Probability that exactly two of those items are defective
Question 12
2 / -0.83

A box contains 20 defective items and 80 non-defective items. If two items are selected at random without replacement, what will be the probability that both items are defective?

A
1/5

B
1/25

C
20/99

D
19/495

Solution

The probability of defective items = 20/100
Therefore the probability of first two defective items without replacement
Question 13
2 / -0.83

If three coins are tossed simultaneously, the probability of getting at least one head is

A
1/8

B
3/8

C
1/2

D
7/8

Solution

Here the sample space S =2³ = 8.
No. of ways to get all tails =1.
∴probability to get all tails = 1/8
∴ Probability to get at least one head is = 1-1/8= 7/8
Question 14
2 / -0.83

If 20 per cent managers are technocrats, the probability that a random committee of 5 managers consists of exactly 2 technocrats is

A
0.2048

B
0.4000

C
0.4096

D
0.9421

Solution
Question 15
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Four arbitrary point (x1,y1), (x2,y2), (x3,y3), (x4,y4), are given in the x, y –plane Using the method of least squares, if, regressing y upon x gives the fitted line y = ax + b; and regressing y upon x given the fitted line y = ax + b; and regressing x upon y gives the fitted line x = cy + d then

A
The two fitted lines must coincide

B
The two fitted lines need not coincide

C
It is possible that ac = 0

D
A must be 1/c

Solution

y =ax+b −(i) and x = cy + d −(ii)
Question 16
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Chebyshev ’s inequality states that the probability of a “Six Sigma ”event is less than :

A
10%

B
20%

C
30%

D
3%

Solution

If a bell curve is assumed, the probability of a “six sigma ”event is on the order of one ten millionth of a percent.
Question 17
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Three values of x and y are to be fitted in a straight line in the form y = a + bx by the method of least squares. Given Σx = 6, Σy = 21, Σx² = 14 and Σxy = 46, the values of a and b are respectively.

A
2 and 3

B
1 and 2

C
2 and 1

D
3 and 2

Solution
Question 18
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A box contains 10 screws, 3 of which are defective. Two screws are drawn at random with replacement. The probability that none of the two screws is defective will be

A
100%

B
50%

C
47%

D
49%

Solution

A = First drawn screw non-defective.

B = Second drawn screw non-defective.

P(A) = 7/10

Because 7 out of 10 screws are non-defective and we sample at random,

P(B) = 7/10

The second drawing is the same as the beginning.

P(A Intersection B) = P(A) * P(B) = 0.7 * 0.7 = 0.49

It is Converted to percentage = 0.49 * 100 = 49%

The probability of two screws are defective is 49%.
Question 19
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Which one of the following statements is NOT true?

A
The measure of skewness is dependent upon the amount of dispersion

B
In a symmetric distribution, the values of mean, mode and median are the same

C
In a positively skewed distribution; mean >median >mode

D
In a negatively skewed distribution; mode >mean >median
Solution
Skewness and Distribution Characteristics
The measure of skewness indicates how much a distribution deviates from being symmetrical.
In a symmetric distribution , the mean, median, and mode are all equal.
A positively skewed distribution has the mean greater than the median, which is greater than the mode (mean > median > mode).
Conversely, in a negatively skewed distribution , the mode is greater than the mean, which is greater than the median (mode > mean > median).
Therefore, the statement claiming that in a negatively skewed distribution, the mode is less than the mean and the mean is less than the median is incorrect.
Question 20
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If the standard deviation of the spot speed of vehicles in a highway is 8.8 kmph and the mean speed of the vehicles is 33 kmph, the coefficient of variation in speed is

A
0.1517

B
0.1867

C
0.2666

D
0.3645

Solution