Comprehensive Probability Foundations And Distribution Analysis Practice Tests

Question 1

An investment portfolio contains 12 stocks, 8 bonds, and 5 cash equivalents. If one asset is selected at random, what is the probability that it is either a stock or a bond?

Accepted Answer

0.8

Answer

0.32

Answer

0.48

Answer

0.2

Question 2

A credit agency observes that in a sample of 500 loans, 15 resulted in default and 35 resulted in late payments. Based on the empirical approach, calculate the probability that a randomly selected loan from this sample is neither in default nor late.

Accepted Answer

0.9

Question 3

Given two events $A$ and $B$ such that $P(A) = 0.4$, $P(B) = 0.3$, and $P(A \cap B) = 0.1$, calculate the probability of the union $P(A \cup B)$.

Accepted Answer

0.6

Answer

0.7

Answer

0.5

Answer

0.12

Question 4

If two events $A$ and $B$ are independent, then the conditional probability $P(A|B)$ must be equal to $P(A)$.

Accepted Answer

True

Answer

False

Question 5

A firm has two independent projects, $X$ and $Y$. The probability of project $X$ succeeding is $0.7$ and the probability of project $Y$ succeeding is $0.6$. What is the probability that both projects succeed?

Accepted Answer

0.42

Answer

1.3

Answer

0.1

Answer

0.88

Question 6

If $P(A \cap B) = 0.15$ and $P(B) = 0.25$, calculate the conditional probability $P(A|B)$.

Accepted Answer

0.6

Question 7

A quality control inspector examines 4 components selected at random from a production line where the probability of a defective component is $p = 0.10$. Let $X$ be the number of defective components. Calculate $P(X = 2)$.

Accepted Answer

0.0486

Answer

0.0081

Answer

0.2916

Answer

0.0656

Question 8

A bank receives an average of $\lambda = 3$ fraudulent transaction reports per day. Calculate the probability that the bank receives exactly 5 reports in a single day using the Poisson distribution.

Accepted Answer

0.1008

Question 9

For a Binomial distribution with $n = 10$ and $p = 0.4$, determine the variance $\sigma^2$.

Accepted Answer

2.4

Answer

4.0

Answer

1.6

Answer

0.24

Question 10

If a random variable $X$ follows a Poisson distribution with mean $\lambda = 2$, then the variance of $X$ is equal to 4.

Accepted Answer

False

Answer

True

Question 11

Which of the following scenarios are best modeled by a discrete random variable? Select all that apply.

Answer

The number of insurance claims filed in a month

Answer

The time elapsed between two consecutive customer arrivals

Answer

The number of defaults in a portfolio of 50 loans

Answer

The annual percentage return on a stock

Question 12

A call center receives an average of 2 calls every 10 minutes. Calculate the expected number of calls ($\lambda$) in a 30-minute interval.

Accepted Answer

6

Question 13

In a Binomial experiment with $n = 5$ and $p = 0.5$, calculate the probability $P(X \geq 1)$.

Accepted Answer

0.96875

Answer

0.03125

Answer

0.5

Answer

0.15625

Question 14

A process follows a normal distribution with mean $\mu = 50$ and standard deviation $\sigma = 10$. If the mean is shifted to $\mu = 60$ while keeping $\sigma$ constant, how does the probability density function curve change?

Accepted Answer

The curve shifts to the right along the horizontal axis without changing its shape.

Answer

The curve becomes flatter and wider.

Answer

The curve becomes taller and narrower.

Answer

The curve shifts to the left along the horizontal axis.

Question 15

A random variable $X$ is normally distributed with $\mu = 120$ and $\sigma = 15$. Calculate the $Z$-score for a value $x = 150$.

Accepted Answer

2

Question 16

Given a standard normal distribution, what is the probability $P(Z \leq 1.96)$?

Accepted Answer

0.975

Answer

0.025

Answer

0.950

Answer

0.997

Question 17

If a dataset is normally distributed, the probability $P(X = \mu)$ is exactly 0.5.

Accepted Answer

True

Answer

False

Question 18

You need to find the value $x$ such that 95% of the distribution lies below it, given $\mu = 100$ and $\sigma = 10$. Which Excel function should be used?

Accepted Answer

NORM.INV(0.95, 100, 10)

Question 19

Which of the following properties are true for the Standard Normal Distribution?

Answer

The mean is 0.

Answer

The standard deviation is 1.

Answer

The total area under the curve is 1.

Answer

The distribution is skewed to the right.

Question 20

A project completion time follows a normal distribution with $\mu = 20$ days and $\sigma = 4$ days. Calculate the probability that the project takes longer than 28 days using the $Z$-score approach.

Accepted Answer

0.0228

Comprehensive Probability Foundations And Distribution Analysis Practice Tests

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Comprehensive Probability Foundations and Distribution Analysis

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