Foundational Probability Concepts
Probability is a branch of mathematics that deals with calculating the likelihood of a given event's occurrence, which is expressed as a number between 1 and 0. Here we explore foundational concepts like events, independence, and various rules that guide the calculation of probabilities in different scenarios.
Basic Definitions
Experiment: A procedure that can be infinitely repeated and has a well-defined set of outcomes.
Sample Space (S): The set of all possible outcomes of an experiment.
Event: A subset of the sample space.
Types of Events
Simple Event: An event with a single outcome.
Compound Event: An event made up of two or more simple events.
Independent and Dependent Events
Independent Events: The occurrence of one event does not affect the probability of the other.
Dependent Events: The occurrence of one event affects the occurrence of another.
Mutually Exclusive Events
Two events are mutually exclusive if they cannot occur at the same time. This means \(P(A \cap B) = 0\).
The Complement Rule
The probability of an event not occurring is 1 minus the probability that it does occur. \(P(A^{c}) = 1 - P(A)\)
Addition Rules
For Mutually Exclusive Events: \(P(A \cup B) = P(A) + P(B)\)
For Non-Mutually Exclusive Events: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
Conditional Probability
The probability of an event given that another event has occurred. \(P(A | B) = \frac{P(A \cap B)}{P(B)}\)
Multiplication Rules
For Independent Events: \(P(A \text{ and } B) = P(A) \cdot P(B)\)
For Dependent Events: \(P(A \text{ and } B) = P(A|B) \cdot P(B)\)
Bayes' Theorem
Describes the probability of an event, based on prior knowledge of conditions that might be related to the event. \(P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}\)
Random Variables
A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. There are two types: discrete and continuous.
Expected Value
The expected value of a random variable gives a measure of the center of the distribution of the variable. Essentially a weighted average, \(E(X) = \sum x_i \cdot P(x_i)\) for discrete variables, or an integral for continuous variables.
Probability Distribution
A probability distribution describes how probabilities are distributed over the values of the random variable. Key distributions include Binomial, Normal, and Poisson.
Binomial Distribution
Used for a finite number of trials, each with the same probability of success. The formula is \(P(X = k) = {n \choose k} p^k (1-p)^{n-k}\), where \(n\) is the number of trials, \(k\) is the number of successes, and \(p\) is the probability of success on a single trial.
Normal Distribution
Describes a symmetrical, bell-shaped curve that is defined by its mean (\(\mu\)) and standard deviation (\(\sigma\)). The probability density function is \(f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}\).
Poisson Distribution
Expresses the probability of a given number of events happening in a fixed interval of time or space, given the average number of times the event happens over that interval. \(P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}\), where \(\lambda\) is the mean number of successes in the given interval.
Z-Scores and the Standard Normal Distribution
A Z-score is a measure of how many standard deviations an element is from the mean. The standard normal distribution is a normal distribution with \(\mu = 0\) and \(\sigma = 1\). The Z-score is calculated by \(Z = \frac{X - \mu}{\sigma}\).
Central Limit Theorem
The Central Limit Theorem states that the distribution of the sum (or average) of a large number of independent, identically distributed variables will be approximately normal, regardless of the underlying distribution.