Probability, Random Variables, and Probability Distributions

A two-way table (contingency table) shows counts or proportions for two categorical variables at the same time. Side-by-side bar charts, segmented bar charts and mosaic plots let you compare one variable across the categories of the other and decide whether the two variables are associated.

From a two-way table you can find joint relative frequencies (a cell ÷ the table total), marginal relative frequencies (a row or column total ÷ the table total) and conditional relative frequencies (a cell ÷ its own row or column total). If the conditional distributions differ from group to group, that's evidence the variables are associated.

A simulation uses a chance device, like random numbers, to imitate a random process, so you can estimate a probability from the relative frequency of the simulated outcomes. The law of large numbers says that over more and more independent trials, the relative frequency of an event settles closer and closer to one value, its probability.

The sample space is the list of all possible outcomes, and its total probability is 1. When outcomes are equally likely, P(E) = (number of outcomes in E) ÷ (total number of outcomes); every probability is between 0 and 1, and the complement rule says P(not E) = 1 − P(E).

Two events are mutually exclusive (disjoint) when they can't happen at the same time, so their joint probability is P(A ∩ B) = 0. You use that fact to justify whether or not two events are mutually exclusive.

A conditional probability, P(A | B), is the chance that A happens given that B has happened: P(A | B) = P(A ∩ B) ÷ P(B). Rearranged, it gives the general multiplication rule, P(A ∩ B) = P(A) · P(B | A), which you'll use with two-way tables and tree diagrams.

Events A and B are independent when knowing that one happened doesn't change the probability of the other, so P(A | B) = P(A) and P(A ∩ B) = P(A) · P(B). The probability that A or B (or both) happens is the union, P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

A random variable gives a number to each outcome of a random process. A discrete probability distribution lists every possible value with its probability (the probabilities add up to 1), and a cumulative distribution gives the probability of being less than or equal to each value.

The mean, or expected value, of a discrete random variable is μX = Σ xᵢ · P(xᵢ), its long-run average value, and its standard deviation, σX = √[Σ (xᵢ − μX)² · P(xᵢ)], measures how far values typically fall from that mean. Both are parameters, and you interpret them in context.

A binomial random variable counts the successes in n independent trials, where each trial has only two outcomes (success or failure) and the same probability of success, p. You find binomial probabilities with P(X = x) = C(n, x) · pˣ · (1 − p)ⁿ⁻ˣ or with technology, and the mean and standard deviation are μX = np and σX = √(np(1 − p)).

A normal distribution is a continuous, unimodal, symmetric, bell-shaped curve set by its mean μ and standard deviation σ. The empirical rule says about 68%, 95% and 99.7% of values fall within 1, 2 and 3 standard deviations of the mean, and you find other areas and percentiles with z-scores and a table or with technology.

A sampling distribution shows the values a statistic, such as a sample mean, takes across all possible samples of the same size; you can approximate one by simulating many samples, or build a randomization distribution by repeatedly reshuffling responses between treatment groups. The central limit theorem says that when your random sample is large enough, the possible values of x̄ pile up in a roughly normal shape even if the population isn't normal, and that shape gets closer to normal as n grows.