"Probability theory or probability calculus is the branch of mathematics concerned with probability."
Introduction to the concept of probability and how it is calculated.
Sample Space: A collection of all possible outcomes of a random experiment.
Events: A subset of the sample space, consisting of the outcomes that we are interested in.
Probability: A measure of the likelihood of an event occurring. It ranges from 0 to 1.
Types of Probability: There are various types of probability such as Empirical Probability, Theoretical Probability, and Conditional Probability.
Complement of an Event: The complement of an event A is the set of all outcomes in which A does not occur.
Mutually Exclusive Events: Two or more events are mutually exclusive if they cannot happen at the same time.
Independent Events: Two or more events are independent if the occurrence of one event does not affect the probability of the other.
Addition Formula for Probability: The addition formula for probability is used to calculate the probability of the union of two or more events.
Multiplication Formula for Probability: The multiplication formula for probability is used to calculate the probability of the intersection of two or more events.
Bayes’ Theorem: Bayes’ theorem is used to update the probability of an event based on new information or evidence.
Expected Value: The expected value is the mean or average value of a random variable.
Variance and Standard Deviation: The variance and standard deviation are measures of the spread or dispersion of a random variable.
Probability Distributions: Probability distributions describe the probabilities of different outcomes in a random experiment.
Discrete Probability Distributions: Discrete probability distributions are probability distributions for discrete random variables.
Continuous Probability Distributions: Continuous probability distributions are probability distributions for continuous random variables.
Probability Density Function: The probability density function is a function that describes the probability distribution of a continuous random variable.
Cumulative Distribution Function: The cumulative distribution function is a function that describes the probability distribution of a random variable up to a certain point.
Normal Distribution: The normal distribution is a continuous probability distribution that is commonly used in statistics and probability.
Central Limit Theorem: The Central Limit Theorem is a theorem that states that the sum or average of a large number of independent and identically distributed random variables will tend to follow a normal distribution.
Hypothesis Testing: Hypothesis testing is a statistical technique that is used to test a claim or hypothesis about a population based on data from a sample.
Classical Probability: This is the simplest form of probability where the possible outcomes are equally likely and easily determined.
Empirical Probability: This is also known as experimental or relative frequency probability. It is based on observation and experimentation of past events.
Subjective Probability: This type of probability is based on an individual’s personal judgement and experience. It is often used when there is no previous data or information available.
Conditional Probability: This type of probability is used to determine the probability of an event occurring given that another event has already occurred.
Joint Probability: This type of probability is used to calculate the probability of two or more events happening simultaneously.
Marginal Probability: This is the probability of an event occurring without taking into consideration other events.
Mutual Exclusive Probability: When two events are mutually exclusive, it means that they cannot happen at the same time. The probability of one event occurring precludes the occurrence of the other event.
Independent Probability: When two events are independent of each other, the occurrence of one event does not affect the probability of the other event occurring.
Tree Diagrams: Tree diagrams are often used to help understand the possible outcomes of a series of events.
Bayes Theorem: Bayes theorem is used to update prior probabilities based on new observations or data.
"Probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms."
"Typically these axioms formalize probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space."
"Any specified subset of the sample space is called an event."
"Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes."
"Stochastic processes provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion."
"Two major results in probability theory describing such behavior are the law of large numbers and the central limit theorem."
"It is not possible to perfectly predict random events."
"As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data."
"Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics or sequential estimation."
"A great discovery of twentieth-century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics."
"...expressing it through a set of axioms."
"...a measure taking values between 0 and 1."
"A set of outcomes called the sample space."
"Discrete and continuous random variables..."
"...mathematical abstractions of non-deterministic or uncertain processes or measured quantities..."
"The law of large numbers describes the behavior of random events."
"The central limit theorem describes the behavior of random events."
"Probability theory is essential to many human activities that involve quantitative analysis of data."
"Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state."