Glossary

Comparing two variants' sample proportions to decide which performs better, a comparison whose reliability depends on each proportion's own standard error.

P(A or B) equals P(A) plus P(B) minus their overlap.

Ignoring how rare a condition actually is when interpreting a positive test result, even though both the rate and the test's accuracy feed into the posterior.

A formula that reverses a conditional probability -- given the evidence, it finds the probability of the cause that produced it.

A formula for reversing a conditional probability, P(A|B) from P(B|A), used throughout this chapter's Monty Hall and Naive Bayes sections.

A single yes/no trial -- success with probability p, failure otherwise. Every indicator random variable is one.

The count of successes across n independent Bernoulli trials, each with the same success probability.

The surprising result that a shared birthday becomes more likely than not among just 23 people, driven by the number of PAIRS being compared.

The technique for finding a transformed variable's own distribution by rewriting its CDF purely in terms of the original variable's own known CDF.

For large n, a sample average is approximately normal, no matter what the original population's own distribution looks like.

A bound on tail probability that holds for every distribution with a finite variance, needing no assumption about the distribution's own shape.

The skill of recognizing which counting technique a word problem actually needs, before any formula gets applied.

An arrangement around a circle, where rotations count as the same arrangement.

The "favorable over total" formula, valid only when every outcome is equally likely.

A selection of items where order does not matter.

The probability of an event not occurring is 1 minus the probability it does occur.

Counting the outcomes that fail a condition, then subtracting from the total -- often far easier than counting the condition directly.

A joint distribution narrowed to one row or column, then rescaled to sum to 1 again.

The mean of a conditional distribution -- what Y averages to, given that X took a specific value.

The probability of an event once the sample space has been restricted to another event that's already known to have occurred.

A hidden third factor that drives both of two correlated variables, producing a strong correlation with no direct causal link between them.

The half-unit shift applied when approximating a discrete binomial probability with a continuous normal curve.

The running total of probability up to x for a continuous random variable, built from an integral instead of a sum, with no jumps anywhere.

Every value in an interval, equally likely -- a perfectly flat density.

Covariance rescaled by both standard deviations into a unitless number between -1 and 1, measuring the strength of a LINEAR relationship.

A counting strategy for constraints like "two specific items together" or "never together" -- fix the restriction in place first, then count what's still free.

A measure of how two random variables move together -- positive when they tend to rise and fall in tandem, negative when one runs high as the other runs low.

The running total of probability at or below a value -- a step function that jumps by p(x) at each value X can take.

The complement of a union is the intersection of the complements, and vice versa.

Every integer value in a fixed range, equally likely -- the simplest named distribution in this course.

The observed relative frequency of an event from actually running an experiment.

For any normal distribution, about 68%, 95%, and 99.7% of values fall within one, two, and three standard deviations of the mean.

A subset of the sample space -- a collection of outcomes sharing some property.

The probability-weighted average of every value a random variable can take -- the long-run average outcome across many repetitions.

The waiting time until the next event of a constant-rate process -- the continuous sibling of the geometric distribution.

The factor that shrinks the standard error when sampling without replacement from a finite population, always at most 1.

The combined name for the product rule and sum rule together, as the two building blocks behind every counting technique.

Wrongly believing independent trials owe a correction after a streak.

The mistaken belief that a past streak of outcomes changes an independent trial's own future probability -- independent trials have no memory.

The probability a gambler with a fixed starting bankroll reaches a target before going broke, playing a fixed-probability game repeatedly.

The number of independent trials up to and including the first success.

A sum where each term is a fixed ratio times the one before it.

The number of successes drawn without replacement from a finite population -- what replaces the binomial when independence fails.

A definite integral with an infinite bound, or an integrand that blows up somewhere in the interval.

A method for finding the probability of a union by adding individual probabilities, then correcting for double-counted overlaps.

X and Y are independent when their joint distribution factors into the product of both marginals, for every single pair.

Two events where one occurring doesn't change the other's probability at all.

A random variable that is 1 when a chosen event occurs and 0 otherwise -- its expected value is just that event's own probability.

Writing a hard-to-analyze count as a sum of simple 0/1 indicator variables, then finding its expected value with linearity instead of its whole distribution.

The distribution of two random variables together, on the same sample space -- pmf for a discrete pair, pdf for a continuous one.

As trials grow, empirical frequency converges toward the theoretical probability.

The formula for the expected value of a function of a random variable, computed directly from that variable's own pmf with no new distribution needed.

A way to compute a probability by summing its conditional probability across every case of a partition, weighted by how likely each case is.

The factor odds form multiplies into the prior odds -- how much more likely the evidence is under one hypothesis than under its complement.

The value a function's outputs settle toward as its input grows without bound.

A random variable rule of the form Y=aX+b -- a stretch (or flip) by a, followed by a slide by b, the only transformation this course solves in full closed form.

Expected value distributes over sums and scaling exactly like ordinary algebra, whether or not the variables involved are independent.

One variable's own distribution, recovered from a joint table by summing (or integrating) the other variable away.

A geometric wait's remaining trials, given no success yet, have exactly the same distribution as a fresh wait -- past failures carry no information.

An exponential wait forgets entirely how long it has already run -- the continuous-time twin of the geometric distribution's own memoryless property.

If A is a subset of B, A can never be more probable than B.

Estimating an unknown quantity by simulating many independent trials and averaging the results, guaranteed to converge by the weak law of large numbers.

A game-show puzzle where switching doors after a non-random reveal doubles the win probability from 1/3 to 2/3.

A generalization of the binomial coefficient to more than two labeled groups.

The conditional probability formula solved for the joint probability -- probability of A, times probability of B given A.

Two events that share no outcomes -- they can never both occur on the same run.

A classifier that applies Bayes' theorem while assuming features are conditionally independent given the class, an assumption rarely exactly true but often good enough to classify well.

The number of independent trials up to and including the r-th success -- r independent geometric waits stacked together.

The familiar bell curve -- symmetric, defined by nothing more than a mean and a standard deviation.

The ratio of an event's probability to its complement's probability -- not a probability itself, and not bounded by 1.

A single element of a sample space.

Checking every pair of several events for independence is a strictly weaker guarantee than checking that all of them are independent together.

A parameter numerically summarizes a population (like mu); a statistic numerically summarizes a sample (like x-bar), used to estimate the corresponding parameter.

A triangular array of binomial coefficients, where each entry is the sum of the two entries above it.

An arrangement of items where order matters.

The single question that decides which formula applies -- does swapping two chosen items count as a different outcome?

The count of distinct arrangements of a set that contains repeated (indistinguishable) items.

A count of events over a fixed interval at a constant average rate lambda -- the one family whose mean and variance are always equal.

A population is the entire group of interest; a sample is the observed subset actually drawn from it.

The three named roles in Bayes' theorem -- the belief before evidence, how probable the evidence is under that belief, and the updated belief after.

The three rules -- nonnegativity, normalization, additivity -- every probability measure must satisfy.

A continuous random variable's density -- its height is not a probability, but the AREA under it over an interval is.

A function listing each value a discrete random variable can take alongside that value's own probability.

A function assigning a number to every event, obeying the three axioms.

A branching picture of a sequential experiment, with each edge labeled by its own conditional probability and each leaf's probability equal to the product along its path.

Probability reasons forward, from a known population to a sample; statistics reasons backward, from an observed sample to an unknown population.

If a choice is made in stages, the total count is the product of each stage's option count.

Treating a small P(evidence given innocent) as though it were the (often much larger) P(innocent given evidence) -- a transposed conditional applied to a courtroom.

The value below which a given proportion of a distribution falls -- the median is the quantile at proportion 0.5.

The fraction of time a single server is busy, rho = lambda/mu -- a queue is stable only when this is below 1.

A function from a sample space's outcomes to numbers -- it turns "which outcome happened" into "what number resulted."

A sequence of positions built by taking independent, identically distributed steps -- gambler's ruin is the bounded special case with absorbing barriers.

The set of every possible outcome of an experiment.

The distribution of a sample statistic (like a sample mean or proportion) across every possible sample of a given size, the bridge probability builds toward statistics.

Shorthand for writing a sum of many similar terms without writing every one out.

The (positive) square root of variance -- a spread measure back in the random variable's own original units.

The standard deviation of the sampling distribution of the mean, sigma over the square root of n -- a different quantity from sigma itself, which never shrinks with n.

A technique for counting distributions of identical items into labeled bins.

If a choice is made from disjoint groups, the total count is the sum of each group's option count.

The probability a system of components stays up, computed differently for components wired in series, in parallel, or in a k-of-n arrangement.

A probability computed from a model -- axioms, counting, or a named distribution.

The expected squared deviation from the mean -- a measure of how spread out a random variable's values are.

As sample size grows, the probability that a sample average deviates from the true mean by more than any fixed amount shrinks toward 0.

How many standard deviations a value sits above or below its distribution's mean -- the conversion that makes different normal distributions directly comparable.