Probability for Statistics and Data Science

Explore how combinatorics relate to probability by examining permutations, variations, and combinations, with or without repetition and order restrictions, and learn to count favorable outcomes and sample spaces.

Explore permutations by counting the distinct arrangements of a set of elements. Compute permutations as the product n times (n-1) times ... times 1, i.e., n factorial.

Explore factorials, defined as the product from 1 to n, with the rule n! = (n-1)!·n. Note that zero factorial equals one, and negative numbers do not have factorials.

Explore variations with repetition and how to count them using the rule v(n,p) = n^p, illustrated by a three-letter code example with a, b, and c.

Explore variations without repetition by arranging p elements from n, illustrated with a five-member relay. Learn the formula n!/(n-p)!, and apply it to count orders when no one runs twice.

Explore combinations and how they avoid double counting by comparing them to variations and permutations. Use the formula n!/(p!(n-p)!) with 3 out of 10 and 4 out of 10 examples.

Explore the symmetry in combinations by showing that selecting p elements from n equals selecting n minus p elements to leave behind, enabling simpler calculations.

Multiply the options for each component to count all possible combinations of events with separate sample spaces, such as lunch menus and online ads.

Calculate the lottery probability by multiplying independent events: five numbers from 69 without repetition and the Powerball from 1–26, giving about 1 in 300 million for a single ticket.

Learn to distinguish permutations, variations, and combinations—with and without repetition—and apply their formulas P=n!, V=n!/(n-p)!, C=n!/(p!(n-p)!). Recognize symmetry and set up for multiple-event probability and Bayesian notation.

Apply combinatorics to a pizza menu by multiplying spaces. Use combinations without repetition, variations with repetition, and combinations with repetition to compare deals and reveal how small changes expand options.

Explore how events are described by sets, including favorable outcomes, element membership, and subset relations, using notation such as ∈, not ∈, for all, such that, and empty sets.

Represent sets and events with circles to show disjoint, intersecting, and subset relationships; use card examples like diamonds, hearts, queens, and red cards to illustrate possible outcomes.

Explore the intersection of events A and B, where both must occur. See examples: hearts and diamonds (empty set), diamonds and queens (queen of diamonds), and red cards with diamonds.

Learn how the union of events A and B counts outcomes satisfying either condition, using the formula union equals A plus B minus their intersection. Explore disjoint and subset cases.

Explore mutually exclusive sets with no overlaps and an empty intersection; their union is the sum of the sets, and complements are the rest of the sample space.

Explore how probabilities shift between independent and dependent events, using conditional probability P(A|B) to illustrate updated chances when new information narrows the sample space.

Explain the difference between P(A|B) and P(B|A) and define conditional probability, illustrating independence and dependence with coin flips and queen-of-spades examples.

Examine conditional probability through a real-life survey, compare p(a|b) and p(b|a), and apply the additive law of total probability with the multiplication rule.

Apply the additive law to compute a union as sum minus intersection. See 0.76 for women or vegetarians and 0.17 for SQL and Tableau, with the multiplication rule next.

Explore the multiplication rule from the conditional probability formula P(A|B) and P(B), with card-drawing examples. Learn to compute the intersection using these probabilities and adjusted sample spaces.

Explore how Bayes theorem links conditional probabilities, including A given B and B given A, with real-world medical and business examples and independence cases.

Apply bayesian inference to real-world college data, using the Hamilton College 2017–2018 common data set to assess diversity among freshmen by gender and race.

Explore what a probability distribution is, its mean and variance, and how standard deviation, population versus sample data, and probability functions describe how outcomes occur.

Compare discrete and continuous distributions, where discrete handles finite outcomes with equiprobable, Bernoulli, and binomial models, while continuous covers infinite outcomes via normal, t, chi-squared, exponential, and logistic families.

Explore discrete distributions and their characteristics, express the distribution with a table or graph, compute interval probabilities by summing outcomes, and note P(Y ≤ y) = P(Y < y+1).

Explore the discrete uniform distribution, where all outcomes have equal probability, illustrated by a six-sided die; learn why the mean and variance offer no predictive power.

Explore the Bernoulli distribution: a single-trial two-outcome model with success probability p, yielding expected value p or 1-p and variance p(1-p), and its binomial link.

Explore how the binomial distribution models n Bernoulli trials using combinations and p, relate it to a single Bernoulli trial, and derive the expected value and variance with real-world examples.

Learn the Poisson distribution with parameter lambda, its discrete probability function P(Y)=lambda^y e^{-lambda}/y!, and that the mean and variance both equal lambda, illustrated with seven questions in a workday.

Explore how continuous distributions differ from discrete ones, with infinite sample space, pdfs, and the relationship between the pdf and cdf through integration and differentiation.

Explore the normal distribution N(mu, sigma squared), a bell-shaped, symmetric curve where data cluster around the mean. Learn its key properties, including the 68, 95, 99.7 law, variance, and standardization.

Learn how standardizing converts any normal distribution to a standard normal distribution by centering at zero and scaling to unit variance, using z-scores and the z-score table.

Understand the student's t distribution, defined by degrees of freedom, as a small-sample alternative to the normal distribution with fatter tails. Apply it to hypothesis testing with limited data.

Explore the chi squared distribution and its three degrees of freedom, and how it underpins hypothesis testing, confidence intervals, and goodness-of-fit for categorical data.

Explore the exponential distribution and its rate parameter lambda, with pdf and cdf behavior. Learn that its mean is 1/lambda, variance 1/lambda^2, and how taking the natural logarithm enables regression.

Explore the continuous logistic distribution, defined by its mean (location) and scale, and how its pdf and cdf shape binary-outcome probabilities in sports forecasting.

Explore how normal, Student's t, Poisson, exponential, and binomial distributions shape real-world data using FIFA 19 stats, viewership, and member trends to guide balance and marketing.

Explore how probability drives finance through option pricing, premium decisions, and payoffs using a decision tree and expected value to decide when to exercise.

See how statistics uses probability to analyze samples and infer population characteristics. Learn how confidence intervals and hypothesis testing quantify uncertainty and how distributions underpin supervised machine learning modeling.

Explore how probability underpins statistics and data science, including Monte Carlo simulations, expected values, and predictive modeling in real-world uncertainty.