Q: Given $P(B) = \frac{3}{12}$ and $P(C) = \frac{8}{12}$, are B and C independent if $P(B \cap C) = \frac{3}{12}$?

No, because $P(B \cap C) \neq P(B) \times P(C)$

Q: A survey asked consumers to rate a detergent as Bad, Average, Good, or Very Good. If 85 out of 212 consumers rated Brand A as Good or Very Good, what percentage is this?

$\frac{85}{212} \times 100 \approx 40.1\%$

Q: How many different passwords can be formed by choosing 3 letters (A-Z, repetition allowed) followed by 2 digits (0-9, repetition allowed)?

$26^3 \times 10^2 = 17,576 \times 100 = 1,581,840$

Question 1

Suppose a box contains 4 white balls and 3 black balls. If you randomly select one ball, what is the probability of drawing a black ball?

Accepted Answer

$\frac{3}{7}$

Question 2

Events A and B are said to be incompatible if:

Accepted Answer

They cannot occur at the same time, i.e., $A \cap B = \emptyset$

Question 3

Given $P(B) = \frac{3}{12}$ and $P(C) = \frac{8}{12}$, are B and C independent if $P(B \cap C) = \frac{3}{12}$?

Accepted Answer

No, because $P(B \cap C) 
eq P(B) 	imes P(C)$

Question 4

A store sells an average of 5.1 items per 8-hour day. Assuming the number of items sold follows a Poisson distribution, what is the probability that more than 2 items are sold in a day?

Accepted Answer

$1 - (e^{-5.1} + 5.1 e^{-5.1} + \frac{5.1^2}{2} e^{-5.1})$

Question 5

If the average number of items sold in 2 hours is 1.275, what is the variance of the number of items sold in 2 hours, assuming a Poisson distribution?

Accepted Answer

$1.275$

Question 6

A survey asked consumers to rate a detergent as Bad, Average, Good, or Very Good. If 85 out of 212 consumers rated Brand A as Good or Very Good, what percentage is this?

Accepted Answer

$\frac{85}{212} 	imes 100 \approx 40.1\%$

Question 7

Given the data set: 18, 21, 12, 17, 34, 13, 12, 26, 25, 31, what is the interquartile range (IQR)?

Accepted Answer

$13$

Question 8

The annual returns for Stock A over 7 years are: 14.3%, 19.0%, 16.4%, 8.9%, 10.3%, 5.1%, 8.8%. What is the coefficient of variation (CV) for Stock A? (Use $CV = \frac{\sigma}{\mu} 	imes 100\%$)

Accepted Answer

Approximately $32.6\%$

Question 9

If the median of a data set is the average of the 12th and 13th values, and these are 366 and 369, what is the median?

Accepted Answer

$\frac{366 + 369}{2} = 367.5$

Question 10

A class of 35 students has an average grade of 70. If the 20 male students have an average grade of 73, what is the average grade of the 15 female students?

Accepted Answer

$66$

Question 11

How many different passwords can be formed by choosing 3 letters (A-Z, repetition allowed) followed by 2 digits (0-9, repetition allowed)?

Accepted Answer

$26^3 	imes 10^2 = 17,576 	imes 100 = 1,581,840$

Question 12

Suppose you can draw a ball from either urn #1 or urn #2. The expected gain from urn #1 is $3933.33 and from urn #2 is $4000. Which urn should you choose to maximize your expected gain?

Accepted Answer

Urn #2, because its expected gain is higher

Question 13

If the expected value of the gain from urn #2 is $4000 and the expected value of the square of the gain is $16,500,000, what is the variance of the gain?

Accepted Answer

$16,500,000 - (4000)^2 = 500,000$

Question 14

If $X$ is a Poisson random variable with mean $\lambda = 5.1$, what is $P(X = 2)$?

Accepted Answer

$\frac{5.1^2}{2!} e^{-5.1}$

Question 15

A police officer records the number of tickets given by each of 10 officers: 18, 21, 12, 17, 34, 13, 12, 26, 25, 31. What is the first quartile (Q1) of this data set?

Accepted Answer

$13$

Question 16

If the largest value in a data set is 424, and the median is the average of the 12th and 13th values (366 and 369), by how much can the largest value be increased without affecting the median?

Accepted Answer

It can be increased by any amount

Question 17

If $P(A) = 0.4$, $P(B) = 0.5$, and $P(A \cap B) = 0.2$, what is $P(A \cup B)$?

Accepted Answer

$0.4 + 0.5 - 0.2 = 0.7$

Question 18

A Poisson process has a mean of 1.275 items per 2 hours. What is the probability of selling at most 1 item in 2 hours?

Accepted Answer

$e^{-1.275} + 1.275 e^{-1.275}$

Probability and Statistics: Exam Review Solutions (MQT-1102)

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