Mathematics

Probability Distributions

457 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

A store sells an average of 20 items per day. What is the probability that the store sells exactly 25 items on a particular day?

  1. 0.135

  2. 0.271

  3. 0.406

  4. 0.542
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let X be the number of items sold on a particular day. Then X follows a Poisson distribution with a mean of 20. The probability of selling exactly 25 items is given by P(X = 25) = (20^25 * e^(-20)) / 25! = 0.271.

Multiple choice

A company receives an average of 50 emails per hour. What is the probability that the company receives no emails in the next 15 minutes?

  1. 0.067

  2. 0.135

  3. 0.203

  4. 0.271
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let X be the number of emails received in the next 15 minutes. Then X follows a Poisson distribution with a mean of 12.5 (50 * 15 / 60). The probability of receiving no emails is given by P(X = 0) = e^(-12.5) = 0.203.

Multiple choice

A doctor sees an average of 10 patients per day. What is the probability that the doctor sees at least one patient on a particular day?

  1. 0.995

  2. 0.950

  3. 0.900

  4. 0.850
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let X be the number of patients seen on a particular day. Then X follows a Poisson distribution with a mean of 10. The probability of seeing at least one patient is given by P(X >= 1) = 1 - P(X = 0) = 1 - e^(-10) = 0.995.

Multiple choice

A machine produces bolts with a probability of 0.99 of being defective. What is the probability that the machine produces at most 2 defective bolts in the next 100 bolts?

  1. 0.904

  2. 0.950

  3. 0.990

  4. 0.999
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let X be the number of defective bolts produced in the next 100 bolts. Then X follows a Poisson distribution with a mean of 99 (0.99 * 100). The probability of producing at most 2 defective bolts is given by P(X <= 2) = P(X = 0) + P(X = 1) + P(X = 2) = e^(-99) + 99 * e^(-99) + (99^2 * e^(-99)) / 2! = 0.990.

Multiple choice

A company receives an average of 20 phone calls per hour. What is the probability that the company receives between 15 and 25 phone calls in the next hour?

  1. 0.242

  2. 0.368

  3. 0.494

  4. 0.620
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let X be the number of phone calls received in the next hour. Then X follows a Poisson distribution with a mean of 20. The probability of receiving between 15 and 25 phone calls is given by P(15 <= X <= 25) = P(X = 15) + P(X = 16) + ... + P(X = 25) = 0.494.

Multiple choice

A store sells an average of 50 items per day. What is the probability that the store sells more than 60 items on a particular day?

  1. 0.067

  2. 0.135

  3. 0.203

  4. 0.271
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let X be the number of items sold on a particular day. Then X follows a Poisson distribution with a mean of 50. The probability of selling more than 60 items is given by P(X > 60) = 1 - P(X <= 60) = 1 - (P(X = 0) + P(X = 1) + ... + P(X = 60)) = 0.203.

Multiple choice

A doctor sees an average of 15 patients per day. What is the probability that the doctor sees exactly 20 patients on a particular day?

  1. 0.102

  2. 0.204

  3. 0.306

  4. 0.408
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let X be the number of patients seen on a particular day. Then X follows a Poisson distribution with a mean of 15. The probability of seeing exactly 20 patients is given by P(X = 20) = (15^20 * e^(-15)) / 20! = 0.204.

Multiple choice

What is the probability of getting a z-score of 1.96 or higher in a standard normal distribution?

  1. 0.025

  2. 0.05

  3. 0.01

  4. 0.005
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The z-score of 1.96 corresponds to the 97.5th percentile in a standard normal distribution. So, the probability of getting a z-score of 1.96 or higher is 0.025.

Multiple choice

What is the probability of getting a sample proportion of 0.5 or higher from a population with a proportion of 0.5, if the sample size is 100?

  1. 0.5

  2. 0.68

  3. 0.95

  4. 0.99
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The sampling distribution of the sample proportion is a normal distribution with a mean equal to the population proportion and a standard deviation equal to the square root of the population proportion times the square root of 1 minus the population proportion, divided by the square root of the sample size. In this case, the sampling distribution of the sample proportion has a mean of 0.5 and a standard deviation of sqrt(0.5 * 0.5) / sqrt(100) = 0.05. So, the probability of getting a sample proportion of 0.5 or higher is 0.99.

Multiple choice

What is the probability of getting a chi-square statistic of 10 or higher from a chi-square distribution with 5 degrees of freedom?

  1. 0.05

  2. 0.10

  3. 0.15

  4. 0.20
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The chi-square distribution with 5 degrees of freedom has a mean of 5 and a standard deviation of sqrt(2 * 5) = sqrt(10). So, the probability of getting a chi-square statistic of 10 or higher is 0.05.

Multiple choice

What is the probability of getting a t-statistic of 2 or higher from a t-distribution with 10 degrees of freedom?

  1. 0.05

  2. 0.10

  3. 0.15

  4. 0.20
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The t-distribution with 10 degrees of freedom has a mean of 0 and a standard deviation of sqrt(10 / (10 - 2)) = sqrt(10 / 8) = sqrt(5 / 4). So, the probability of getting a t-statistic of 2 or higher is 0.05.

Multiple choice

What is the probability of getting an F-statistic of 4 or higher from an F-distribution with 5 and 10 degrees of freedom?

  1. 0.05

  2. 0.10

  3. 0.15

  4. 0.20
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The F-distribution with 5 and 10 degrees of freedom has a mean of 5 and a standard deviation of sqrt(2 * 5 * 10 / (5 - 2)) = sqrt(100 / 3) = sqrt(100 / 3). So, the probability of getting an F-statistic of 4 or higher is 0.05.

Multiple choice

What is the probability of getting a correlation coefficient of 0.5 or higher from a population with a correlation coefficient of 0.5?

  1. 0.05

  2. 0.10

  3. 0.15

  4. 0.20
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The sampling distribution of the correlation coefficient is a t-distribution with n-2 degrees of freedom, where n is the sample size. In this case, the sampling distribution of the correlation coefficient has 100-2 = 98 degrees of freedom. The probability of getting a correlation coefficient of 0.5 or higher is 0.10.

Multiple choice

What is the probability of getting a regression coefficient of 2 or higher from a population with a regression coefficient of 2?

  1. 0.05

  2. 0.10

  3. 0.15

  4. 0.20
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sampling distribution of the regression coefficient is a t-distribution with n-2 degrees of freedom, where n is the sample size. In this case, the sampling distribution of the regression coefficient has 100-2 = 98 degrees of freedom. The probability of getting a regression coefficient of 2 or higher is 0.05.

Multiple choice

What is the probability of getting an ANOVA F-statistic of 4 or higher from a population with an ANOVA F-statistic of 4?

  1. 0.05

  2. 0.10

  3. 0.15

  4. 0.20
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sampling distribution of the ANOVA F-statistic is an F-distribution with k-1 and n-k degrees of freedom, where k is the number of groups and n is the total sample size. In this case, the sampling distribution of the ANOVA F-statistic has 2-1 = 1 and 100-2 = 98 degrees of freedom. The probability of getting an ANOVA F-statistic of 4 or higher is 0.05.