Probability Cheat Sheet

The core ideas of Probability distilled into a single, scannable reference — perfect for review or quick lookup.

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Quick Reference

Sample Space and Events

The sample space is the set of all possible outcomes of a random experiment, while an event is any subset of the sample space. Defining these precisely is the first step in any probability analysis, since probabilities are assigned to events rather than to individual outcomes in many frameworks.

Conditional Probability

Conditional probability measures the likelihood of an event occurring given that another event has already occurred. It is defined as $P(A|B) = \frac{P(A \cap B)}{P(B)}$, provided $P(B) > 0$. This concept is essential for updating beliefs when new information becomes available.

Bayes' Theorem

Bayes' theorem provides a formula for reversing conditional probabilities: $P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$. It allows us to update a prior belief about event $A$ after observing evidence $B$, forming the foundation of Bayesian inference and decision-making under uncertainty.

Law of Large Numbers

The Law of Large Numbers states that as the number of independent, identically distributed trials increases, the sample average converges to the expected value. This theorem explains why casinos are profitable in the long run and why polling works despite individual unpredictability.

Central Limit Theorem

The Central Limit Theorem states that the sum or average of a large number of independent random variables, regardless of their original distribution, tends toward a normal (Gaussian) distribution. This result justifies the widespread use of normal-distribution-based methods in statistics.

Random Variables and Distributions

A random variable is a function that assigns a numerical value to each outcome in a sample space. Its probability distribution describes the likelihood of each possible value. Distributions can be discrete (e.g., binomial, Poisson) or continuous (e.g., normal, exponential).

Expected Value and Variance

The expected value (mean) of a random variable is the long-run average of its outcomes, weighted by their probabilities. Variance measures how spread out values are around the mean. Together, they summarize a distribution's center and dispersion.

Independence and Dependence

Two events are independent if the occurrence of one does not affect the probability of the other, meaning $P(A \cap B) = P(A) \cdot P(B)$. When events are dependent, their joint probability requires conditioning. Recognizing independence is critical for simplifying calculations and building correct models.

Combinatorics in Probability

Combinatorics provides counting techniques such as permutations, combinations, and the multiplication principle that are essential for computing probabilities in finite sample spaces. These methods determine how many ways events can occur, which directly feeds into probability calculations.

Bayesian vs. Frequentist Interpretation

The frequentist interpretation defines probability as the long-run relative frequency of an event in repeated experiments. The Bayesian interpretation treats probability as a degree of belief, updated with evidence via Bayes' theorem. These paradigms lead to different statistical methodologies.

Key Terms at a Glance

Sample Space:The set of all possible outcomes of a random experiment, typically denoted by $S$ or $\Omega$.

Event:A subset of the sample space representing one or more outcomes of interest.

Probability:A numerical measure between 0 and 1 that quantifies the likelihood of an event occurring.

Conditional Probability:The probability of an event $A$ given that event $B$ has occurred, calculated as $P(A|B) = \frac{P(A \cap B)}{P(B)}$.

Bayes' Theorem:A formula that relates conditional probabilities: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$, used to update beliefs with new evidence.

Random Variable:A function that assigns a numerical value to each outcome in a sample space.

Expected Value:The weighted average of all possible values of a random variable, where weights are the probabilities of each value.

Variance:A measure of how spread out the values of a random variable are around the mean, equal to $E[(X - \mu)^2]$.

Standard Deviation:The square root of the variance, providing a measure of spread in the same units as the random variable.

Independence:Two events are independent when the occurrence of one does not affect the probability of the other: $P(A \cap B) = P(A)P(B)$.

Mutual Exclusivity:Two events are mutually exclusive if they cannot both occur simultaneously, meaning $P(A \cap B) = 0$.

Binomial Distribution:A discrete probability distribution describing the number of successes in $n$ independent Bernoulli trials with success probability $p$.

Normal Distribution:A continuous probability distribution with a symmetric bell-shaped curve, defined by mean $\mu$ and standard deviation $\sigma$.

Poisson Distribution:A discrete distribution that models the number of events occurring in a fixed interval at a constant average rate $\lambda$.

Central Limit Theorem:A theorem stating that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the population distribution.

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