- "Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true."
The study of the likelihood of events occurring and the principles governing their occurrence, such as conditional probability and Bayes' theorem.
Sets and Set Operations: Understanding the concept of sets and operations like union, intersection, and complement are important in probability theory.
Probability Spaces: This involves understanding the concept of sample space, events, and probability measure for those events.
Conditional Probability: This involves understanding the probability of an event given that another event has already occurred.
Bayes’ Theorem: A formula that shows how to update probabilities based on new information.
Random Variables: This involves understanding the concept of a variable that takes on values based on a random process, and its probability distribution.
Probability Distributions: Probability distributions are used to describe the likelihood of different outcomes in a random process.
Joint Probability Distributions: Also known as Multivariate Probability Distributions or Joint Probability Mass Functions (PMFs), they describe the probabilities of all possible combinations of events.
Independence: Events are independent if the probability of one event occurring does not affect the probability of another event occurring.
Expectation and Variance: The expected value is the long-term average value of a random variable, and the variance measures how much the values of a random variable fluctuate.
Conditional Expectation and Variance: These are measures of expected value and variance that take into account additional information.
Law of Large Numbers: This theorem states that as the number of trials gets larger, the observed frequency will approach the true probability.
Central Limit Theorem: This theorem states that as the sample size gets larger, the sample mean will approach a normal distribution.
Statistical Inference: Statistical inference is the process of drawing conclusions about a population from a sample.
Hypothesis Testing: This is a process of testing a hypothesis about a population parameter.
Confidence Intervals: Confidence intervals provide a range of values that the true parameter is likely to lie within.
Regression Analysis: Regression analysis is a statistical method used to investigate relationships between variables.
Classical Probability: It involves finding the ratio of the number of favorable outcomes to the total number of possible outcomes.
Empirical Probability: It is the probability estimated based on observed data, by using the relative frequency of events.
Subjective Probability: This type of probability is assigned based on the belief or expectation of an individual, without any statistical backup.
Conditional Probability: It is the probability of an event occurring given that another event has already occurred.
Joint Probability: The probability of two or more events occurring together, usually calculated using the multiplication rule.
Marginal Probability: The probability of an individual event occurring, without taking into account other events.
Prior Probability: Probability assigned prior to obtaining any new information or data.
Posterior Probability: Probability assigned after obtaining new information or data.
Cumulative Probability: The probability of an event occurring up to a particular point in time, usually calculated using the cumulative distribution function.
Bayes’ Probability: This type of probability is updated based on new evidence or information and is used in Bayesian statistics.
Frequency Probability: It is a type of empirical probability where the probability of an event is estimated based on how often the event has occurred in the past.
Geometric Probability: It deals with the probability of geometric shapes or arrangements, such as points, lines, and circles.
Poisson Probability: It is used to model the probability of rare events occurring, such as accidents or equipment failures.
Bernoulli Probability: The probability of a binary event occurring, such as heads or tails in a coin toss.
Markov Chain Probability: It involves predicting the probability of future events based on the current state, using Markov chains.
Monte Carlo Probability: This type of probability is estimated through a large number of simulations, using random numbers.
Weibull Probability: It is used to model the probability distribution of failure times for complex systems.
Zero Probability: The probability of a particular event occurring is zero.
- "The probability of an event is a number between 0 and 1."
- "The higher the probability of an event, the more likely it is that the event will occur."
- "A simple example is the tossing of a fair (unbiased) coin... the probability of either 'heads' or 'tails' is 1/2 (which could also be written as 0.5 or 50%)."
- "Probability theory is used widely in areas of study such as statistics, mathematics, science, finance, gambling, artificial intelligence, machine learning, computer science, game theory, and philosophy."
- "These concepts have been given an axiomatic mathematical formalization in probability theory."
- "Probability theory is used... to draw inferences about the expected frequency of events."
- "Probability theory is also used to describe the underlying mechanics and regularities of complex systems."
- "0 indicates impossibility of the event and 1 indicates certainty."
- "Since the coin is fair, the two outcomes ('heads' and 'tails') are both equally probable."
- "The probability of 'heads' equals the probability of 'tails'."
- "...the probability of either 'heads' or 'tails' is 1/2 (which could also be written as 0.5 or 50%)."
- "The probability of either 'heads' or 'tails' is 1/2 (which could also be written as 0.5 or 50%)."
- "Probability theory is used widely in... finance."
- "Probability theory is used widely... in machine learning."
- "Probability theory is used widely... in game theory."
- "Probability theory is used widely... in philosophy to, for example, draw inferences about the expected frequency of events."
- "Probability theory is used widely... in artificial intelligence."
- "Probability theory is used to describe the underlying mechanics and regularities of complex systems."
- "0 indicates impossibility of the event."