GATE Overflow for GATE ESE - Recent questions in Probability and Statistics

GATE ES 2026 | Question: 4

Mon, 23 Feb 2026 13:15:42 +0000

A box contains $26$ cards with all English alphabets from $\text{A}$ to $\text{Z}$.

The probability of randomly choosing the card with either '$\text{E}$' or '$\text{S}$' from this box is $\_\_\_\_$.

$2 / 26$
$1 / 26$
$3 / 26$
$4 / 26$

GATE ES 2026 | Question: 5

Mon, 23 Feb 2026 13:15:41 +0000

Which ONE of the following options is the CORRECT formula for Chi-Square $\left(\chi^{2}\right)$ 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 $E_{i}, i=1, \ldots, n$.

$\chi^{2}=\sum_{\mathrm{i}=1}^{\mathrm{n}} \frac{\left(O_{i}-E_{i}\right)^{2}}{E_{i}}$
$\chi^{2}=\sqrt{\frac{\sum_{i=1}^{n}\left(O_{i}-\bar{O}\right)}{n-1}}$
$\chi^{2}=\dfrac{\bar{O}-\mu}{\sigma_{o} \sqrt{n}}$
$\chi^{2}=\dfrac{\left(\sigma_{o}\right)^{2}}{\left(\sigma_{E}\right)^{2}}$

GATE ES 2026 | Question: 24

Mon, 23 Feb 2026 13:15:02 +0000

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).

GATE ES 2026 | Question: 38

Mon, 23 Feb 2026 13:14:36 +0000

Standard deviation for the particulate matter (PM) concentrations listed in the table is $\_\_\_\_$ $\mu \mathrm{g} / \mathrm{m}^{3}$ (rounded off to one decimal place).
$$\begin{array}{|l|c|c|c|c|c|c|c|}
\hline
\textbf{Location No.} & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
\begin{array}{l}
\textbf{PM concentration} \\
(\mu \mathrm{g}/\mathrm{m}^{3})
\end{array}
& 45 & 40 & 55 & 65 & 60 & 35 & 50 \\
\hline
\end{array}$$

GATE ES 2025 | Question: 14

Thu, 06 Mar 2025 16:22:22 +0000

Consider the following statements

Environmental pollutant concentration is generally modeled using lognormal distribution.
Environmental pollutant concentration is generally modeled using Poisson distribution.
The weekly rate of exceedance of environmental pollutant concentration with regards to a given standard is generally modeled using lognormal distribution.
The weekly rate of exceedance of environmental pollutant concentration with regards to a given standard is generally modeled using Poisson distribution.

Choose the correct option(s) from the following

$(i)$ and $(iv)$ are correct
$(i)$ is correct and $(iii)$ is incorrect
$(ii)$ and $(iv)$ are correct
$(ii)$ and $(iv)$ are incorrect

GATE ES 2025 | Question: 37

Thu, 06 Mar 2025 16:21:16 +0000

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)

GATE ES 2024 | GA Question: 7

Mon, 19 Feb 2024 10:29:49 +0000

The probability of a boy or a girl being born is $1 / 2$. For a family having only three children, what is the probability of having two girls and one boy?

$3 / 8$
$1 / 8$
$1 / 4$
$1 / 2$

GATE ES 2024 | Question: 1

Mon, 19 Feb 2024 10:24:14 +0000

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

$4 / 10$
$6 / 10$
$2 / 10$
$3 / 10$

GATE ES 2024 | Question: 26

Mon, 19 Feb 2024 10:24:09 +0000

A population $\text{(with mean $\mu$)}$ 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, $\mathrm{C}(\mathrm{t})$ of the cumulative probability function for Student's $t$ - distribution two-tailed test with degree of freedom, $\text{D}$

Which one of the following expression is correct for testing the null hypothesis $H_{0}: \mu=0$ at $10 \%$ significance level?

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

$-1.81<\frac{x}{\frac{S}{\sqrt{N-1}}}<1.81$
$-1.83<\frac{x}{\frac{S}{\sqrt{N-1}}}<1.83$
$-1.37<\frac{x}{\frac{S}{\sqrt{N-1}}}<1.37$
$-2.23<\frac{x}{\frac{S}{\sqrt{N-1}}}<2.23$

GATE ES 2023 | GA Question: 3

Tue, 23 May 2023 03:15:39 +0000

Given a fair six-faced dice where the faces are labelled $'1'$, $'2'$, $'3'$, $'4'$, $'5'$, and $'6'$, what is the probability of getting a $'1'$ on the first roll of the dice and a $'4'$ on the second roll?

$\frac{1}{36}$
$\frac{1}{6}$
$\frac{5}{6}$
$\frac{1}{3}$

GATE ES 2023 | Question: 20

Tue, 23 May 2023 03:12:23 +0000

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).

GATE ES 2023 | Question: 26

Tue, 23 May 2023 03:12:18 +0000

$\text{W1, W2, W3...W9}$ represent the holding times of $9$ water samples, which follow a normal distribution with mean $=8.33$ and standard deviation $=4.472$.$\text{M}$ represents the sample mean value of holding times, which also has a normal distribution. Assuming $\mathrm{Z}$ has a standard normal distribution $\text{(mean $=0$ and standard deviation $=1$)}$, select the correct statement which describes the expression for calculating the value of type $1$ error where
$$
\begin{array}{ll}
\text { null hypothesis }\left(\mathrm{H}_0\right): & \mathrm{M}>6 \\
\text { alternate hypothesis }\left(\mathrm{H}_{\mathrm{a}}\right): & \mathrm{M} \leq 6
\end{array}
$$

$\mathrm{P}\{\mathrm{Z}<(-1.565)\}$
$\mathrm{P}\{\mathrm{Z}<1.565\}$
$\mathrm{P}\{\mathrm{Z}>(-1.565)\}$
$\mathrm{P}\{\mathrm{Z}>1.565\}$

GATE ES 2021 | Question: 27

Tue, 14 Mar 2023 03:44:23 +0000

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?

$\frac{35}{91}$
$\frac{56}{91}$
$\frac{12}{21}$
$\frac{9}{21}$

GATE ES 2022 | Question: 1

Fri, 17 Feb 2023 07:06:16 +0000

A student reported following arsenic concentrations in water samples:
$$\begin{array}{|l|l|l|l|l|l|l|l|}\hline \text{Arsenic concentration (mg/L)} & 0.10 & 0.12 & 0.20 & 0.05 & 0.40 & 0.30 & 0.35 \\\hline \end{array}$$
Which one of the following is a correct statement about arsenic concentration distribution?

Arsenic concentration distribution is symmetric
Arsenic concentration distribution is positively skewed
Arsenic concentration distribution is negatively skewed
Arsenic concentration is following Weibull distribution

GATE ES 2022 | Question: 2

Fri, 17 Feb 2023 07:06:15 +0000

$\mathrm{X}$ is normally distributed with following data $(25.8,36.6,26.3,21.8,27.2)$. Select the correct statement about $X\left(\mathrm{t}_{\text {crit }, \alpha}=0.05,4=2.132\right)$ :

Population mean $\leq 25$ with $95 \%$ confidence
Population mean $\leq 25$ with $100 \%$ confidence
Population mean $>25$ with $95 \%$ confidence
Population mean $>25$ with $100 \%$ confidence

GATE ES 2022 | Question: 6

Fri, 17 Feb 2023 07:06:13 +0000

Ten litres of the sample was filtered through a membrane filter and the filter was transferred to solid agar media which supports growth of coliform organisms. After incubation at $37{ }^{\circ} \mathrm{C}$ for $48$ hours, the agar plates showed an average of $64$ colonies per plate. What was the average concentration of coliform organisms (in Colony Forming Units per unit volume) in the original water sample?

$64 \;\mathrm{CFU} / \mathrm{ml}$
$640 \; \mathrm{CFU} / \mathrm{ml}$
$6.4 \times 10^{-3} \; \mathrm{CFU} / \mathrm{ml}$
$64 \times 10^{-3} \; \mathrm{CFU} / \mathrm{ml}$

GATE ES 2022 | Question: 43

Fri, 17 Feb 2023 07:05:41 +0000

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