Chance looks messy one trial at a time and orderly over thousands. This unit starts with two-way tables, builds the rules of probability, turns outcomes into random variables with means and standard deviations, and meets the two distributions the rest of the course runs on: the binomial and the normal. It ends with why sample means behave so predictably.
A two-way table counts two categorical variables at once. Joint, marginal and conditional relative frequencies answer different questions, and comparing conditional relative frequencies across groups (often with a segmented bar chart) shows whether the variables are associated.
A random process has outcomes decided by chance. The probability of an event is its long-run relative frequency, and the law of large numbers says that with more independent trials, the relative frequency gets closer to that value. Simulation estimates probabilities by modeling the random process.
The sample space lists every possible outcome and has probability 1. With equally likely outcomes, P(E) is favorable outcomes over total outcomes. P(not E) = 1 − P(E). Events are mutually exclusive when they can't happen together, so their joint probability is 0.
P(A | B) is the probability of A when B is known to have happened. The general multiplication rule gives P(A ∩ B). Events are independent when knowing one happened doesn't change the other's probability. The addition rule subtracts the overlap so it isn't counted twice.
A random variable assigns a number to each outcome of a random process. Its probability distribution lists every value with its probability, summing to 1. The expected value is the long-run average, and the standard deviation is the typical distance of values from it over the long run.
A binomial random variable counts successes in a fixed number of independent trials that each succeed with the same probability. Its mean is np and its standard deviation is √(np(1 − p)), and the binomial probability function gives P(X = x).
A normal distribution is a continuous, unimodal, symmetric bell curve set by its mean μ and standard deviation σ. Areas under the curve are probabilities: the empirical rule gives quick estimates, and z-scores with technology or a table give exact areas and the boundaries for a given area.
A sampling distribution is the distribution of a statistic over all possible samples of a given size. Simulation approximates it, and a randomization distribution does the same for reassigned treatment groups. The central limit theorem says the sampling distribution of a sample mean is approximately normal, more so for larger samples.