Probability and Statistics - Mathematics Foundation - Environmental Science & Engineering Previous Year Questions

1

2024 · Environmental Science & Engineering · Mathematics Foundation · Probability and Statistics

Environmental Science & Engineering (ES) 2024

Ten cards in a pack are numbered as 1, 2, 3. . . .10. The probability of drawing a card with an even number or a number which is a multiple of 5 from the pack is _____.

A

4/10

B

6/10

C

2/10

D

3/10

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2

2024 · Environmental Science & Engineering · Mathematics Foundation · Probability and Statistics

Environmental Science & Engineering (ES) 2024

A population (with mean μ) follows normal distribution. Ten samples (N) are drawn at random with a mean value of “x” and standard deviation of “S”. Following table provides the confidence limits, C(t) of the cumulative probability function for Student’s t – distribution two-tailed test with degree of freedom, D.

Which one of the following expression is correct for testing the null hypothesis H₀: μ = 0 at 10% significance level?

D	C(t)
	0.9	0.95	0.975
9	1.38	1.83	2.26
10	1.37	1.81	2.23
11	1.36	1.80	2.20

A

\(-1.81 < \frac{x}{\frac{S}{\sqrt{N-1}}} < 1.81 \)

B

\(-1.83 < \frac{x}{\frac{S}{\sqrt{N-1}}} < 1.83 \)

C

\(-1.37 < \frac{x}{\frac{S}{\sqrt{N-1}}} < 1.37 \)

D

\(-2.23 < \frac{x}{\frac{S}{\sqrt{N-1}}} < 2.23 \)

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3

2025 · Environmental Science & Engineering · Mathematics Foundation · Probability and Statistics

Environmental Science & Engineering (ES) 2025

A residential family is considering two cities for relocation. The data related to pollutant exposure and associated health cost per year are given in the following figure. The pollutant exposure is characterized in high, mild and low exposure categories with respective probability values. The difference in expected value of health cost of City 1 with respect to that of City 2 is __________ lakhs/year. (rounded off to two decimal places).

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4

2021 · Environmental Science & Engineering · Mathematics Foundation · Probability and Statistics

Environmental Science & Engineering (ES) 2021

The table below shows the carbon content of four samples of powdered coal. If these four samples are mixed completely, what would be the resultant carbon percentage of the mixture by weight?

Sample number	Mass of sample (kg)	Carbon % by weight
1	2	88
2	1	90
3	2	80
4	1	90

A

58 %

B

86 %

C

87 %

D

90 %

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5

2021 · Environmental Science & Engineering · Mathematics Foundation · Probability and Statistics

Environmental Science & Engineering (ES) 2021

A biased die has six faces numbered as \(k = 1, 2, 3, 4, 5,\) and 6. On rolling the die, the probability of the number \(k\) appearing is proportional to \(k^2\). What is the probability that an even number will appear on rolling the die?

A

\(\frac{35}{91}\)

B

\(\frac{56}{91}\)

C

\(\frac{12}{21}\)

D

\(\frac{9}{21}\)

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6

2022 · Environmental Science & Engineering · Mathematics Foundation · Probability and Statistics

Environmental Science & Engineering (ES) 2022

A student reported following arsenic concentrations in water samples:

Arsenic concentration (mg/L)	0.10	0.12	0.20	0.05	0.40	0.30	0.35

Which one of the following is a correct statement about arsenic concentration distribution?

A

Arsenic concentration distribution is symmetric

B

Arsenic concentration distribution is positively skewed

C

Arsenic concentration distribution is negatively skewed

D

Arsenic concentration is following Weibull distribution

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7

2022 · Environmental Science & Engineering · Mathematics Foundation · Probability and Statistics

Environmental Science & Engineering (ES) 2022

X is normally distributed with following data (25.8, 36.6, 26.3, 21.8, 27.2). Select the correct statement about X (t_{crit, α=0.05, 4} = 2.132):

A

Population mean ≤ 25 with 95% confidence

B

Population mean ≤ 25 with 100% confidence

C

Population mean > 25 with 95% confidence

D

Population mean > 25 with 100% confidence

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8

2022 · Environmental Science & Engineering · Mathematics Foundation · Probability and Statistics

Environmental Science & Engineering (ES) 2022

An individual has four different email accounts, 60% of emails come into his corporate account, 30% come into his gmail account and remaining 10% are equally divided into his yahoo and zoho accounts. Only 1% of the emails in his corporate accounts are spam, whereas corresponding percentages for gmail, yahoo and zoho accounts are 2%, 3% and 3%, respectively. Assuming that the same spam filter is used in all the four email accounts, the probability (in percentage, rounded off to one decimal place) of having a randomly selected email as spam is __________

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9

2023 · Environmental Science & Engineering · Mathematics Foundation · Probability and Statistics

Environmental Science & Engineering (ES) 2023

A researcher compiled the following information about the performance of a kit in an outbreak

Infection state	Kit response
Disease (probability = 0.002)	Positive response (probability = 0.98)
No Disease	Positive response (probability = 0.03)

The probability of detecting an infection for a positive result through the kit would be __________ (rounded off to three decimal places).

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10

2023 · Environmental Science & Engineering · Mathematics Foundation · Probability and Statistics

Environmental Science & Engineering (ES) 2023

W1, W2, W3…W9 represent the holding times of 9 water samples, which follow a normal distribution with mean = 8.33 and standard deviation = 4.472. M represents the sample mean value of holding times, which also has a normal distribution. Assuming Z has a standard normal distribution (mean = 0 and standard deviation = 1), select the correct statement which describes the expression for calculating the value of type 1 error where
null hypothesis (H₀): M > 6
alternate hypothesis (H₁): M ≤ 6

A

P{Z<(–1.565)}

B

P{Z<1.565}

C

P{Z>(-1.565)}

D

P{Z>1.565}

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11

2026 · Environmental Science & Engineering · Mathematics Foundation · Probability and Statistics

Environmental Science & Engineering (ES) 2026

A box contains 26 cards with all English alphabets from A to Z.
The probability of randomly choosing the card with either ‘E’ or ‘S’ from this box is ____.

A

2/26

B

1/26

C

3/26

D

4/26

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12

2026 · Environmental Science & Engineering · Mathematics Foundation · Probability and Statistics

Environmental Science & Engineering (ES) 2026

Which ONE of the following options is the CORRECT formula for Chi-Square (\(\chi^2\)) test?
Given, \(O_1, O_2, \ldots, O_n\) are the observed values, \(E_1, E_2, \ldots, E_n\) are the corresponding expected values, \(\bar{O}\) is the mean of observed values, \(n\) is the number of observations, \(\mu\) is the mean of \(E_i, i = 1, \ldots, n\), \(\sigma_o\) is the standard deviation of observed values, and \(\sigma_E\) is the standard deviation of expected values \(E_i, i = 1, \ldots, n\).

A

\(\chi^2 = \sum_{i=1}^{n} \frac{(O_i - E_i)^2}{E_i}\)

B

\(\chi^2 = \sqrt{\frac{\sum_{i=1}^{n}(O_i - \bar{O})}{n-1}}\)

C

\(\chi^2 = \frac{\bar{O} - \mu}{\sigma_o/\sqrt{n}}\)

D

\(\chi^2 = \frac{(\sigma_o)^2}{(\sigma_E)^2}\)

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13

2026 · Environmental Science & Engineering · Mathematics Foundation · Probability and Statistics

Environmental Science & Engineering (ES) 2026

A 6-hour rainfall of 6 cm at a station was found to have a return period of 40 years.
The probability that a 6-hour rainfall of this or larger magnitude will occur at least once in 20 successive years at that station is ______ (rounded off to three decimal places).

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14

2026 · Environmental Science & Engineering · Mathematics Foundation · Probability and Statistics

Environmental Science & Engineering (ES) 2026

Standard deviation for the particulate matter (PM) concentrations listed in the table is ______ µg/m³ (rounded off to one decimal place).

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