Glossary

Glossary of probability & statistics

Short, plain-language definitions of the concepts we keep coming back to across the blog — from probability and the binomial distribution to random walks and the Central Limit Theorem.

Bernoulli trial: A single random experiment with exactly two possible outcomes, usually called success and failure. A single coin flip is the textbook example: heads or tails, with a fixed probability for each. String many independent Bernoulli trials together and you build up richer ideas like the binomial distribution, which counts the successes across a whole batch of flips.
Binomial coefficient: A count of how many ways you can choose k items from a set of n, when the order does not matter. Read aloud as "n choose k", it answers questions like how many different ways there are to get exactly k heads in n coin flips. For 10 flips, there are 252 ways to land exactly five heads, which is why that outcome is the most common.
Binomial distribution: The probability distribution of the number of successes in a fixed number of independent yes/no trials, each with the same probability of success. The classic example is the number of heads in 10 fair coin flips: getting zero or ten heads is rare, while counts near five are most likely. Its shape is symmetric and bell-like when the success probability is one half.
Brownian motion: The random, jittery movement of tiny particles suspended in a fluid, caused by constant collisions with the surrounding molecules. Mathematically it is the continuous-time limit of a random walk: shrink the coin-flip steps smaller and smaller and take more and more of them, and the jagged path smooths into Brownian motion. It underpins models of diffusion and financial markets.
Catalan numbers: A famous sequence of whole numbers — 1, 1, 2, 5, 14, 42 and so on — that counts a surprising range of balanced or recursive structures. They appear in coin-flip and ballot problems, for instance counting paths of heads and tails that never let one side fall behind the other. They turn up naturally in random walks and in many counting puzzles across combinatorics.
Central Limit Theorem: A cornerstone result saying that when you add up or average many independent random variables, the result tends toward a normal, bell-shaped distribution — no matter what the original distribution looked like. It is why the count of heads over many coin flips piles up symmetrically around the expected value and traces out a smooth bell curve as the number of flips grows.
Combinatorics: The branch of mathematics concerned with counting, arranging and combining things. It is the toolkit behind probability: before you can say how likely a pattern of coin flips is, you often have to count how many ways it can happen. Combinatorics answers questions like how many distinct sequences of heads and tails exist, or how many of them contain exactly three heads.
Exponent: A small raised number that tells you how many times to multiply a base by itself. Two with an exponent of three means two times two times two, which is eight. Exponents explain why possibilities explode with coin flips: the number of equally likely outcomes of n flips is two to the power n, so ten flips give 1024 distinct head-and-tail sequences.
Pascal's triangle: A triangular array of numbers where each entry is the sum of the two directly above it, starting from a single 1 at the top. Each row lists the binomial coefficients, which count the head-and-tail combinations for a given number of flips: row four, for example, reads 1, 4, 6, 4, 1 — the number of ways to get zero, one, two, three or four heads in four coin flips.
Probability: A number between 0 and 1 that measures how likely an event is to happen. 0 means impossible, 1 means certain, and 0.5 (or 50%) means equally likely either way — like a single fair coin landing on heads. It can be read as a long-run frequency: flip a fair coin many times and roughly half the tosses come up heads.
Probability distribution: A description that assigns a probability to each possible outcome of a random process, with all the probabilities adding up to 1. For a single fair coin flip the distribution is simply heads 0.5 and tails 0.5. For richer setups, like the number of heads in many flips, it spreads probability across many outcomes and reveals which results are common and which are rare.
Random walk: A path built from a sequence of random steps. A simple example uses a coin: step right on heads, left on tails, and watch where you end up after many flips. Even though each step is fair, the walker can wander far from the start. Random walks model everything from diffusing particles to stock prices, and they connect directly to coin-flip probability.
Standard deviation: A measure of how spread out a set of values is around their average. A small standard deviation means most values sit close to the mean; a large one means they are widely scattered. For coin flips it tells you how far the number of heads typically strays from the expected count — for example, in 100 fair flips the head count usually lands within about 5 of fifty.
The gambler's fallacy: The mistaken belief that past independent results change the odds of what comes next. After a run of heads, it feels like tails is "due" — but a fair coin has no memory, so every flip stays an even 50/50. Streaks happen by chance and do not make the opposite outcome more likely on the next toss.
The law of large numbers: As you repeat a random trial more and more times, the observed average gets closer to the expected value. Flip a fair coin many times and the proportion of heads tends toward 0.5. Importantly, it does not "balance out" short runs: a streak of heads is never corrected by future flips. Only the long-run proportion settles down, while the raw count of heads can keep drifting.