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

A company has two production lines, A and B. Line A produces 70% of the company's output, and Line B produces the remaining 30%. Line A has a 5% defect rate, while Line B has a 2% defect rate. If a randomly selected product is defective, what is the probability that it came from Line B?

  1. 0.3

  2. 0.7

  3. 0.2

  4. 0.5
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To calculate the probability that a defective product came from Line B, we need to consider the proportion of output from each line and their respective defect rates. The probability can be calculated as follows: P(Defective from Line B) = P(Line B and Defective) / P(Defective) = (0.3 * 0.02) / (0.7 * 0.05 + 0.3 * 0.02) = 0.006 / 0.008 = 0.2.

Multiple choice

A survey found that 40% of people prefer coffee, 30% prefer tea, and 20% prefer both coffee and tea. What is the probability that a randomly selected person prefers only tea?

  1. 0.1

  2. 0.2

  3. 0.3

  4. 0.4
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To calculate the probability that a person prefers only tea, we need to subtract the proportion of people who prefer both coffee and tea from the proportion of people who prefer tea. The probability can be calculated as follows: P(Prefers Only Tea) = P(Prefers Tea) - P(Prefers Both) = 0.3 - 0.2 = 0.1.

Multiple choice

A medical test has a 98% sensitivity and a 90% specificity. If the prevalence of a disease in a population is 2%, what is the probability that a randomly selected person who tests positive actually has the disease?

  1. 0.98

  2. 0.90

  3. 0.86

  4. 0.14
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To calculate the probability that a person who tests positive actually has the disease, we need to consider the sensitivity and specificity of the test, as well as the prevalence of the disease. The probability can be calculated using Bayes' Theorem as follows: P(Disease|Positive Test) = P(Positive Test|Disease) * P(Disease) / P(Positive Test) = 0.98 * 0.02 / (0.98 * 0.02 + 0.1 * 0.98) = 0.86.

Multiple choice

In a 95% confidence interval, what is the probability that the true population parameter falls within the interval?

  1. 95%

  2. 99%

  3. 90%

  4. 80%
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The level of confidence indicates the probability that the true population parameter falls within the interval. In a 95% confidence interval, there is a 95% chance that the true population parameter is captured within the interval.

Multiple choice

What is the formula for calculating the attributable risk?

  1. Attributable Risk = (Incidence of disease in exposed - Incidence of disease in unexposed) / Incidence of disease in exposed

  2. Attributable Risk = (Prevalence of disease in exposed - Prevalence of disease in unexposed) / Prevalence of disease in exposed

  3. Attributable Risk = (Mortality rate in exposed - Mortality rate in unexposed) / Mortality rate in exposed

  4. Attributable Risk = (Odds of disease in exposed - Odds of disease in unexposed) / Odds of disease in exposed
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Attributable Risk is calculated by dividing the difference between the incidence of disease in exposed individuals and the incidence of disease in unexposed individuals by the incidence of disease in exposed individuals.

Multiple choice

In a queueing system, what is the probability that a customer will have to wait for service?

  1. P(W > 0)

  2. P(W < 0)

  3. P(W = 0)

  4. P(W = \infty)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability that a customer will have to wait for service is denoted by P(W > 0), where W is the waiting time.

Multiple choice

In a queueing system, what is the probability that the system is empty?

  1. P(N = 0)

  2. P(N > 0)

  3. P(N < 0)

  4. P(N = \infty)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability that the system is empty is denoted by P(N = 0), where N is the number of customers in the system.

Multiple choice

What is a probability space in Probability Theory?

  1. A set of all possible outcomes of an experiment.

  2. A set of all possible events in an experiment.

  3. A triple consisting of a sample space, a sigma-algebra of events, and a probability measure.

  4. A function that assigns a probability to each event in an experiment.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A probability space is a mathematical model that describes the behavior of a random experiment. It consists of a sample space, which is the set of all possible outcomes of the experiment, a sigma-algebra of events, which is a collection of subsets of the sample space, and a probability measure, which assigns a probability to each event in the sigma-algebra.

Multiple choice

What is the probability of an event in a probability space?

  1. The ratio of the number of favorable outcomes to the total number of possible outcomes.

  2. The ratio of the number of unfavorable outcomes to the total number of possible outcomes.

  3. The difference between the number of favorable outcomes and the number of unfavorable outcomes.

  4. None of the above.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability of an event in a probability space is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes. It is a number between 0 and 1, where 0 indicates that the event is impossible and 1 indicates that the event is certain.

Multiple choice

Which of the following is an example of a probability distribution?

  1. The uniform distribution.

  2. The normal distribution.

  3. The binomial distribution.

  4. All of the above.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Probability distributions are functions that assign probabilities to events in a probability space. Examples of probability distributions include the uniform distribution, which assigns equal probability to all outcomes in a sample space, the normal distribution, which is a bell-shaped curve that is often used to model continuous random variables, and the binomial distribution, which is used to model the number of successes in a sequence of independent experiments.

Multiple choice

The 'Vyavahara Bija Ganita' of Lilavati provides a method for calculating the 'vyavahara bija', which is a measure of the risk associated with a particular hazard. What is the formula for calculating the vyavahara bija?

  1. Vyavahara bija = (Likelihood of occurrence) x (Severity of consequences)

  2. Vyavahara bija = (Likelihood of occurrence) + (Severity of consequences)

  3. Vyavahara bija = (Likelihood of occurrence) / (Severity of consequences)

  4. Vyavahara bija = (Likelihood of occurrence) - (Severity of consequences)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula for calculating the vyavahara bija is Vyavahara bija = (Likelihood of occurrence) x (Severity of consequences). The likelihood of occurrence is a measure of how likely it is that the hazard will occur, while the severity of consequences is a measure of the potential consequences of the hazard occurring.

Multiple choice

Which mathematical concept is used to describe the probability of an event occurring?

  1. Probability

  2. Statistics

  3. Calculus

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

Probability is a mathematical concept that quantifies the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain).

Multiple choice

The probability of selecting any member of the population is the same in:

  1. Simple Random Sampling

  2. Systematic Sampling

  3. Stratified Sampling

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

In simple random sampling, the probability of selecting any member of the population is the same.

Multiple choice

What is the probability of an event occurring if it has a 30% chance of happening?

  1. 0.3

  2. 0.6

  3. 0.7

  4. 0.9
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability of an event occurring is the likelihood of it happening. In this case, the probability of the event occurring is 30%, which is equal to 0.3.

Multiple choice

What is the probability that a randomly selected person in India will travel by bus?

  1. 0.25

  2. 0.5

  3. 0.75

  4. 1
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

According to the Indian Ministry of Road Transport and Highways, approximately 75% of all passenger trips in India are made by bus.