Probability and Statistics - Engineering Mathematics - Petroleum Engineering Previous Year Questions

1

2016 · Petroleum Engineering · Engineering Mathematics · Probability and Statistics

Petroleum Engineering (PE) 2016

A box has a total of ten identical sized balls. Seven of these balls are black in colour and the rest three are red. Three balls are picked from the box one after another without replacement. The probability that two of the balls are black and one is red is equal to __________.

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2

2017 · Petroleum Engineering · Engineering Mathematics · Probability and Statistics

Petroleum Engineering (PE) 2017

The unbiased sample variance for the set of numbers: S = {40,45,50,55,60} is ________. (write answer with one decimal place)

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3

2017 · Petroleum Engineering · Engineering Mathematics · Probability and Statistics

Petroleum Engineering (PE) 2017

The value of constant \(a\) for which: \(f(x) = \begin{cases} ax^2, & 0 \leq x \leq 5 \\ 0, & otherwise \end{cases}\) is a valid probability density function, is (given, \(a \geq 0\)):

A

\(\frac{1}{125}\)

B

\(\frac{3}{125}\)

C

\(\frac{6}{125}\)

D

\(\frac{9}{125}\)

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4

2018 · Petroleum Engineering · Engineering Mathematics · Probability and Statistics

Petroleum Engineering (PE) 2018

The probability density for three binomial distributions (D1, D2, and D3) is plotted against number of successful trials in the given figure.
Each of the plotted distributions corresponds to a unique pair of (n, p) values, where, n is the number of trials and p is the probability of success in a trial. Three sets of (n, p) values are provided in the table.

Set	(n, p)
I	(60, 0.3)
II	(60, 0.2)
III	(24, 0.5)

Pick the correct match between the (n, p) set and the plotted distribution.

A

Set I – D1, Set II – D2, Set III – D3

B

Set I – D3, Set II – D1, Set III – D2

C

Set I – D2, Set II – D3, Set III – D1

D

Set I – D2, Set II – D1, Set III – D3

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5

2018 · Petroleum Engineering · Engineering Mathematics · Probability and Statistics

Petroleum Engineering (PE) 2018

A box contains 100 balls of same size, of which, 25 are black and 75 are white. Out of 25 black balls, 5 have a red dot. A trial consists of randomly picking a ball and putting it back in the same box, i.e., sampling is done with replacement. Two such trials are done. The conditional probability that no black ball with a red dot is picked given that at least one black ball is picked, is ______. (in fraction rounded-off to two decimal places)

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6

2019 · Petroleum Engineering · Engineering Mathematics · Probability and Statistics

Petroleum Engineering (PE) 2019

Match the crystal structure in Column A with the corresponding packing fractions in Column B of the table

Column A	Column B
1 Simple cubic	P 0.74
2 Hexagonal close-packed	Q 0.68
3 Body-centered cubic	R 0.52
4 Face-centered cubic

A

1-P, 2-R, 3-Q, 4-Q

B

1-R, 2-P, 3-R, 4-Q

C

1-R, 2-P, 3-Q, 4-P

D

1-P, 2-R, 3-P, 4-Q

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7

2019 · Petroleum Engineering · Engineering Mathematics · Probability and Statistics

Petroleum Engineering (PE) 2019

The SQC chart based on Binomial distribution is

A

p chart

B

c chart

C

X̄ chart

D

R chart

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8

2019 · Petroleum Engineering · Engineering Mathematics · Probability and Statistics

Petroleum Engineering (PE) 2019

The average proportion non-conforming of 20 samples each of size 100 items is 0.12. The upper control limit for the relevant chart is _____ (round off to 2 decimal places)

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9

2019 · Petroleum Engineering · Engineering Mathematics · Probability and Statistics

Petroleum Engineering (PE) 2019

A warehouse has 1 loading dock and 3 persons for loading operations. The arrival rate of trucks follows Poisson distribution with a mean of 4 trucks/hour. The average loading time (by three persons together) per truck is exponentially distributed with a mean of 10 minutes. The charge of the trucks per hour and loading charges per person per hour are Rs.20 and Rs.6, respectively. The total cost (in Rs./hour) is _____

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10

2019 · Petroleum Engineering · Engineering Mathematics · Probability and Statistics

Petroleum Engineering (PE) 2019

An acceptance sampling plan is selected with sample size \( n = 80 \), acceptance number \( c = 2 \) for a lot size of 10,000 units. The probability of accepting the lot is based on Poisson distribution. Assuming rectification inspection, if incoming lot quality \( p \) is 0.03 and mean (\( \lambda \)) is 2.4, the average outgoing quality (AOQ) is closest to

A

0.0011

B

0.0087

C

0.0170

D

0.0338

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11

2019 · Petroleum Engineering · Engineering Mathematics · Probability and Statistics

Petroleum Engineering (PE) 2019

The mean time to repair (MTTR) for a repairable system is 30 minutes. When maintenance time changes from 20 minutes to 40 minutes, the net increase in maintainability is closest to

A

0.15

B

0.25

C

0.45

D

0.60

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12

2019 · Petroleum Engineering · Engineering Mathematics · Probability and Statistics

Petroleum Engineering (PE) 2019

A process which is in a state of statistical control (within \( \pm 3\sigma \) ) has an estimate of standard deviation (\( \sigma \)) 2 mm. The specification limits for the corresponding product are \( 120 \pm 8 \) mm. When process mean shifts from 118 mm to 122 mm with no change in process standard deviation, the difference in process capability index \( C_{pk} \) is _____

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13

2019 · Petroleum Engineering · Engineering Mathematics · Probability and Statistics

Petroleum Engineering (PE) 2019

A monitoring system has seven components. The reliability of each component is shown in the figure. The system reliability is _____ (round off to 2 decimal places)

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14

2019 · Petroleum Engineering · Engineering Mathematics · Probability and Statistics

Petroleum Engineering (PE) 2019

A PERT project network consists of 5 activities A to E. The time estimates of these activities follow Beta-distribution. The predecessor-successor (P-S) relationships between the nodes and time estimates of activities are given in table.

Activity	P-S	Optimistic time (days)	Most likely time (days)	Pessimistic time (days)
A	1 – 2	2	4	6
B	2 – 3	4	5	12
C	2 – 4	5	8	11
D	3 – 5	2	5	08
E	4 – 5	4	6	14

The variance (in days) of the critical path is _____ (round off to 2 decimal places)

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15

2020 · Petroleum Engineering · Engineering Mathematics · Probability and Statistics

Petroleum Engineering (PE) 2020

P, Q, R and S are to be uniquely coded using α and β. If P is coded as αα and Q as αβ, then R and S, respectively, can be coded as ______.

A

βα and αβ

B

ββ and αα

C

αβ and ββ

D

βα and ββ

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16

2020 · Petroleum Engineering · Engineering Mathematics · Probability and Statistics

Petroleum Engineering (PE) 2020

The bar graph shows the data of the students who appeared and passed in an examination for four schools P, Q, R and S. The average of success rates (in percentage) of these four schools is ______.

A

58.5 %

B

58.8 %

C

59.0 %

D

59.3 %

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17

2020 · Petroleum Engineering · Engineering Mathematics · Probability and Statistics

Petroleum Engineering (PE) 2020

The number of power outages in a city in a given time interval is a Poisson random variable with a mean of 2 power outages per month. The Poisson distribution is given by, \( P(y) = \frac{e^{-\mu}\mu^y}{y!} \).

The probability of exactly 2 power outages in 2 months (rounded off to two decimal places) is ______________.

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18

2020 · Petroleum Engineering · Engineering Mathematics · Probability and Statistics

Petroleum Engineering (PE) 2020

A data set containing \(n (= 10)\) independent measurements \((x_i, y_i)\) is to be fitted by a simple linear regression model. The least square estimates of regression coefficients are obtained and the regression estimate is given by \(\hat{y} = \widehat{\beta_0} + \widehat{\beta_1}x\).

\(\widehat{\beta_0} = \frac{\sum_{i=1}^{n} y_i}{n} - \widehat{\beta_1} \frac{\sum_{i=1}^{n} x_i}{n}\) and \(\widehat{\beta_1} = \frac{\text{Cov}(x,y)}{\text{Var}(x)}\),

where, \(\text{Cov}(x,y)\) is the sample covariance and \(\text{Var}(x)\) is the sample variance.

The values are given below:

\(\sum_{i=1}^{10} x_i = 25,\quad \sum_{i=1}^{10} y_i = 37\)

\(\sum_{i=1}^{10} x_i^2 = 108,\quad \sum_{i=1}^{10} y_i^2 = 155,\quad \sum_{i=1}^{10} x_i y_i = 120\)

For the given data set, the unbiased variance for the error \((y_i - \hat{y}_i)\) (rounded off to two decimal places) is ________.

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19

2021 · Petroleum Engineering · Engineering Mathematics · Probability and Statistics

Petroleum Engineering (PE) 2021

Nostalgia is to anticipation as ________ is to ________
Which one of the following options maintains a similar logical relation in the above sentence?

A

Present, past

B

Future, past

C

Past, future

D

Future, present

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20

2021 · Petroleum Engineering · Engineering Mathematics · Probability and Statistics

Petroleum Engineering (PE) 2021

Consider the following sentences:
(i) I woke up from sleep.
(ii) I woked up from sleep.
(iii) I was woken up from sleep.
(iv) I was wokened up from sleep.
Which of the above sentences are grammatically CORRECT?

A

(i) and (ii)

B

(i) and (iii)

C

(ii) and (iii)

D

(i) and (iv)

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