Probability Theory Basics

Experiment: An action or situation that can produce a set of outcomes
Sample Space: The set of all possible outcomes of an experiment
Event: A subset of the sample space
Probability: A number between 0 and 1 that represents the likelihood of an event occurring

The probability of an event is always between 0 and 1: 0 ≤ P(A) ≤ 1
The probability of the sample space is always 1: P(S) = 1
The probability of the empty set is always 0: P(∅) = 0
The probability of the union of two events is the sum of their individual probabilities minus the probability of their intersection: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Independent Events: The occurrence of one event does not affect the probability of the other event
Dependent Events: The occurrence of one event affects the probability of the other event
Mutually Exclusive Events: Events that cannot occur simultaneously
Exhaustive Events: Events that cover all possible outcomes of the sample space

Discrete Random Variable: A random variable that takes on a countable number of distinct values
Continuous Random Variable: A random variable that takes on an uncountable number of values
Probability Distribution: A function that describes the probability of each possible value of a random variable

Bernoulli Distribution: Models the probability of success in a single trial
Binomial Distribution: Models the probability of success in multiple trials
Uniform Distribution: Models the probability of each value in a continuous interval being equally likely
Normal Distribution: Models the probability of values in a continuous interval following a bell-shaped curve

An experiment is an action or situation that can produce a set of outcomes, such as flipping a coin or rolling a die
A sample space is the set of all possible outcomes of an experiment, like {heads, tails} for a coin flip
An event is a subset of the sample space, such as getting heads on a coin flip
Probability is a number between 0 and 1 that represents the likelihood of an event occurring, where 0 means impossible and 1 means certain

The probability of an event is always between 0 and 1, meaning it's a percentage or proportion
The probability of the sample space is always 1, because one of the outcomes must occur
The probability of the empty set is always 0, because it can't occur
The probability of the union of two events is the sum of their individual probabilities minus the probability of their intersection, like P(A or B) = P(A) + P(B) - P(A and B)

Independent events are events where the occurrence of one doesn't affect the probability of the other, like flipping two separate coins
Dependent events are events where the occurrence of one affects the probability of the other, like drawing cards from a deck
Mutually exclusive events are events that can't occur simultaneously, like getting both heads and tails on a single coin flip
Exhaustive events are events that cover all possible outcomes of the sample space, like getting either heads or tails on a coin flip

The Theorem of Total Probability states that the probability of an event is the sum of the probabilities of the event occurring given different conditions, like P(A) = P(A|B)P(B) + P(A|B')P(B')
Bayes' Theorem states that the probability of an event given a condition is the probability of the condition given the event times the probability of the event, divided by the probability of the condition, like P(A|B) = P(B|A)P(A) / P(B)

A discrete random variable takes on a countable number of distinct values, like the number of heads in 10 coin flips
A continuous random variable takes on an uncountable number of values, like the height of a person
A probability distribution is a function that describes the probability of each possible value of a random variable, like a graph of probabilities

The Bernoulli distribution models the probability of success in a single trial, like getting heads on a coin flip
The binomial distribution models the probability of success in multiple trials, like getting exactly k heads in n coin flips
The uniform distribution models the probability of each value in a continuous interval being equally likely, like rolling a die
The normal distribution models the probability of values in a continuous interval following a bell-shaped curve, like human heights