Mathematics

Probability Distributions

488 Questions

Probability distributions describe the likelihood of different outcomes in a statistical experiment. Key topics include calculating confidence intervals, understanding Type II errors, and working with the standard normal distribution. These concepts are frequently tested in mathematics and statistics sections of various competitive exams.

Standard normal distributionType II error probabilityConfidence interval interpretationPoisson distributionProbability density function

Probability Distributions Questions

Multiple choice

What is the expected value of a random variable X with probability density function f(x) = \frac{1}{2}e^{-|x|}?

  1. 0

  2. 1

  3. 2

  4. 3
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The expected value of a random variable X with probability density function f(x) is defined as E(X) = ∫xf(x)dx. In this case, E(X) = ∫x\frac{1}{2}e^{-|x|}dx = 1.

Multiple choice

Given two independent events A and B with P(A) = 0.4 and P(B) = 0.6, what is the probability of both A and B occurring?

  1. 0.2

  2. 0.24

  3. 0.36

  4. 0.48
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The probability of both A and B occurring is given by P(A ∩ B) = P(A)P(B). Therefore, P(A ∩ B) = 0.4 * 0.6 = 0.24.

Multiple choice

Which of the following is an example of a discrete random variable?

  1. Height of a randomly selected person

  2. Number of heads in 10 coin tosses

  3. Stock prices over time

  4. Number of customers in a store at a given time
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A discrete random variable can take on only a countable number of values. The number of heads in 10 coin tosses is an example of a discrete random variable because it can take on only the values 0, 1, 2, ..., 10.

Multiple choice

What is the expected value of a random variable X with probability mass function P(X = x) = 1/6 for x = 1, 2, 3, 4, 5, and 6?

  1. 3.5

  2. 4

  3. 4.5

  4. 5
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The expected value of a random variable X is defined as E(X) = ΣxP(X = x). In this case, E(X) = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 3.5.

Multiple choice

A survey of 100 people found that 60 people like chocolate, 40 people like vanilla, and 20 people like both chocolate and vanilla. What is the probability that a randomly selected person from the survey likes chocolate or vanilla?

  1. 0.8

  2. 0.6

  3. 0.4

  4. 0.2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let C be the event that a person likes chocolate and V be the event that a person likes vanilla. Then, P(C or V) = P(C) + P(V) - P(C and V) = 60/100 + 40/100 - 20/100 = 80/100 = 0.8.

Multiple choice

A company has a 10% chance of winning a contract. If the company submits 10 bids, what is the probability that they will win at least one contract?

  1. 0.63

  2. 0.74

  3. 0.85

  4. 0.95
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The probability of the company winning at least one contract is 1 - the probability of the company winning no contracts. The probability of the company winning no contracts is (0.9)^10 = 0.3487. Therefore, the probability of the company winning at least one contract is 1 - 0.3487 = 0.95.

Multiple choice

A machine produces defective items with a probability of 0.05. If 100 items are produced, what is the probability that exactly 5 items are defective?

  1. 0.074

  2. 0.135

  3. 0.201

  4. 0.268
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

This is a binomial distribution problem. The probability of exactly 5 defective items is given by the formula P(X = 5) = (100 choose 5) * (0.05)^5 * (0.95)^95 = 0.201.

Multiple choice

A box contains 100 light bulbs, of which 10 are defective. If 10 light bulbs are randomly selected from the box, what is the probability that exactly 2 of them are defective?

  1. 0.252

  2. 0.308

  3. 0.364

  4. 0.420
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

This is a hypergeometric distribution problem. The probability of exactly 2 defective light bulbs is given by the formula P(X = 2) = (10 choose 2) * (10/100)^2 * (90/100)^8 = 0.308.

Multiple choice

A population has a mean of 100 and a standard deviation of 15. If a sample of 100 people is randomly selected from the population, what is the probability that the sample mean will be between 95 and 105?

  1. 0.682

  2. 0.764

  3. 0.841

  4. 0.910
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The sample mean follows a normal distribution with a mean of 100 and a standard deviation of 15/sqrt(100) = 1.5. Therefore, the probability that the sample mean will be between 95 and 105 is P(95 < X < 105) = P((95 - 100)/1.5 < Z < (105 - 100)/1.5) = P(-3.33 < Z < 3.33) = 0.841.

Multiple choice

A company claims that their product will last for at least 100 hours. If the product actually lasts for an average of 120 hours with a standard deviation of 10 hours, what is the probability that a randomly selected product will last for less than 100 hours?

  1. 0.023

  2. 0.067

  3. 0.111

  4. 0.159
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The product lifetime follows a normal distribution with a mean of 120 hours and a standard deviation of 10 hours. Therefore, the probability that a randomly selected product will last for less than 100 hours is P(X < 100) = P((100 - 120)/10 < Z) = P(Z < -2) = 0.023.

Multiple choice

A survey found that 60% of people prefer coffee, 30% prefer tea, and 10% prefer both coffee and tea. If a person is randomly selected from the survey, what is the probability that they prefer coffee or tea?

  1. 0.7

  2. 0.8

  3. 0.9

  4. 1.0
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let C be the event that a person prefers coffee and T be the event that a person prefers tea. Then, P(C or T) = P(C) + P(T) - P(C and T) = 0.6 + 0.3 - 0.1 = 0.8.

Multiple choice

A machine produces defective items with a probability of 0.01. If 1000 items are produced, what is the probability that the number of defective items is between 5 and 10?

  1. 0.242

  2. 0.383

  3. 0.524

  4. 0.665
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

This is a binomial distribution problem. The probability of between 5 and 10 defective items is given by the formula P(5 ≤ X ≤ 10) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) = 0.383.

Multiple choice

A survey found that 40% of people prefer dogs, 30% prefer cats, and 20% prefer both dogs and cats. If a person is randomly selected from the survey, what is the probability that they prefer dogs or cats?

  1. 0.7

  2. 0.8

  3. 0.9

  4. 1.0
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let D be the event that a person prefers dogs and C be the event that a person prefers cats. Then, P(D or C) = P(D) + P(C) - P(D and C) = 0.4 + 0.3 - 0.2 = 0.9.

Multiple choice

What is the birthday paradox?

  1. The probability that two people in a group of 23 or more people have the same birthday is greater than 50%.

  2. The probability that two people in a group of 23 or more people have the same birthday is less than 50%.

  3. The probability that two people in a group of 23 or more people have the same birthday is exactly 50%.

  4. The probability that two people in a group of 23 or more people have the same birthday is unknown.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The birthday paradox states that the probability that two people in a group of 23 or more people have the same birthday is greater than 50%.

Multiple choice

What is the interpretation of a 95% confidence interval?

  1. There is a 95% chance that the true population parameter is within the interval

  2. The probability of obtaining a sample mean within the interval is 95%

  3. The interval contains the true population parameter with a probability of 95%

  4. The sample mean is within the interval with a probability of 95%
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A 95% confidence interval means that if we were to repeatedly take samples from the population and construct confidence intervals, 95% of those intervals would contain the true population parameter.