# Probability and Statistics

 Question 1
A remote village has exactly 1000 vehicles with sequential registration numbers starting from 1000 . Out of the total vehicles, $30 \%$ are without pollution clearance certificate. Further, even- and oddnumbered vehicles are operated on even- and oddnumbered dates, respectively.
If 100 vehicles are chosen at random on an evennumbered date, the number of vehicles expected without pollution clearance certificate is.
 A 15 B 30 C 50 D 70
GATE CE 2023 SET-2   Engineering Mathematics
Question 1 Explanation:
Since $30 \%$ of the total vehicles are without pollution clearance certificate.
Out of the 100 chosen vehicle, $30 \%$ i.e. $100 \times 0.3=30$ vehicle are expected to be without pollution clearance certificate.
 Question 2
Which of the following probability distribution functions (PDFs) has the mean greater than the median?

 A Function 1 B Function 2 C Function 3 D Function 4
GATE CE 2023 SET-2   Engineering Mathematics
Question 2 Explanation:

Option (B) is correct.

 Question 3
The probabilities of occurrences of two independent events $A$ and $B$ are $0.5$ and $0.8$, respectively. What is the probability of occurrence of at least $A$ or $B$ (rounded off to one decimal place)?
 A 0.4 B 0.2 C 0.9 D 0.7
GATE CE 2023 SET-1   Engineering Mathematics
Question 3 Explanation:
\begin{aligned} & P(A)=0.5 \\ & P(B)=0.8 \end{aligned}
Probability of occurence of atleast $A$ or $B=$ $\mathrm{P}(\mathrm{A} \cap \mathrm{B})$
\begin{aligned} P(A \cap B) & =P(A)+P(B)-P(A \cap B) \\ & =0.5+0.8-P(A) \times P(B) \\ & =0.5+0.8-0.5 \times 0.8 \\ & =1.3-0.4 \\ & =0.9 \end{aligned}
 Question 4
A pair of six-faced dice is rolled thrice. The probability that the sum of the outcomes in each roll equals 4 in exactly two of the three attempts is ______. (round off to three decimal places)
 A 0.045 B 0.078 C 0.018 D 0.025
GATE CE 2022 SET-2   Engineering Mathematics
Question 4 Explanation:
Event, E = {(1, 3)(3, 1)(2, 2)}
n(E) = 3
n(S) = 36
$p=P(E)=\frac{3}{36}=\frac{1}{12}$
$q=P(\bar{E})=1-\frac{1}{12}=\frac{11}{12}$
$P(x)=3C_2(p^2)(q^1)=3 \times \left (\frac{1}{12} \right )^2\left (\frac{11}{12} \right )=0.02$
 Question 5
Match the following attributes of a city with the appropriate scale of measurements.
$\begin{array}{|l|l|}\hline \text{Attribute}&\text{Scale of measurement} \\ \hline \text{(P) Average temperature } (^{\circ}C)\text{ of a city} & \text{((I) Interval} \\ \hline \text{(Q) Name of a city} & \text{(II) Ordinal}\\ \hline \text{(R) Population density of a city}& \text{(III) Nominal}\\ \hline \text{(S) Ranking of a city based on ease of business}& \text{(IV) Ratio}\\ \hline \end{array}$
Which one of the following combinations is correct?
 A (P)-(I), (Q)-(III), (R)-(IV), (S)-(II) B (P)-(II), (Q)-(I), (R)-(IV), (S)-(III) C (P)-(II), (Q)-(III), (R)-(IV), (S)-(I) D (P)-(I), (Q)-(II), (R)-(III), (S)-(IV)
GATE CE 2022 SET-2   Engineering Mathematics
Question 5 Explanation:
Meaning of
Nominal -> a name or term
Ordinal -> in an ordered sequence
Ratio -> quantitative relation between two things
Interval -> indicates average of a range

There are 5 questions to complete.