Guide 1 · Probability

Count What Could Happen

Probability is a disciplined way of listing what could happen and weighing each possibility. Get the list right and most puzzles become arithmetic; get it wrong and no formula will rescue you.

Two dice resting on green felt, photographed close with a shallow depth of field — Illustrative photograph — not measured evidence.

Start with the sample space

A sample space is the complete list of outcomes an experiment can produce. An event is any collection of those outcomes. A probability is a number from 0 to 1 attached to each event: 0 means impossible, 1 means certain.

Modern probability rests on three rules set out by Andrey Kolmogorov in 1933: probabilities are never negative, the whole sample space has probability 1, and for events that cannot happen together, probabilities add. Everything else — conditional probability, Bayes' rule, expected value — follows from those three.

When every outcome is equally likely, probability reduces to counting: favorable outcomes divided by possible outcomes. That shortcut only works when the "equally likely" assumption is true. A weighted die, a biased survey or a sports match does not give you equal outcomes, and treating them as if it did is the first common mistake.

Worked example: the sum of two dice

Two fair six-sided dice give 6 × 6 = 36 equally likely ordered outcomes. A sum of 7 can be made six ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), so its probability is 6/36 = 1/6. A sum of 2 needs double ones: 1/36.

Number of ways (out of 36) to obtain each sum. These are exact theoretical counts for fair dice, not results from rolling real dice.

Conditional probability and Bayes' rule

New information shrinks the sample space. If you know a die roll is even, the possible outcomes are {2, 4, 6}, so the probability it is a 6 rises from 1/6 to 1/3. That is conditional probability: P(6 | even) = P(6 and even) ÷ P(even) = (1/6) ÷ (1/2) = 1/3.

Bayes' rule lets you reverse the direction of a condition. Tests tell us P(positive | condition), but a patient wants P(condition | positive). Our home page example shows how a 99%-sensitive test can still produce mostly false positives when the condition is rare.

The Monty Hall problem

Behind three doors are one car and two goats. You pick a door; the host, who knows where the car is, always opens another door showing a goat and offers a switch. Switching wins with probability 2/3. Your first pick is right 1/3 of the time, and the host's forced choice means that in the other 2/3 of cases the car sits behind the remaining door. The answer depends on the host's rules: if the host opened a door at random and happened to reveal a goat, switching would win only half the time. Assumptions about how the information was produced matter as much as the information itself.

Independence and the gambler's fallacy

Two events are independent when knowing one tells you nothing about the other: P(A | B) = P(A). Successive flips of a fair coin are independent, so after five heads the next flip is still 50/50.

The law of large numbers says the proportion of heads settles toward 1/2 over many flips. It does not say that short-run imbalances get "corrected". They are simply diluted by the many flips that follow.

Surprising coincidences

In a group of 23 people, the probability that at least two share a birthday is about 50.7% (assuming 365 equally likely birthdays and ignoring leap years). It feels too high because we picture matches with our own birthday, while the question counts all 253 possible pairs. Coincidences become likely when there are many chances for one to happen.

Expected value and responsible decisions

The expected value of a bet is the average result per play over a very long run: each outcome's payoff multiplied by its probability, then summed.

On a European roulette wheel there are 37 pockets. A 1-unit bet on a single number pays 35 units if it wins:

E = 35 × (1/37) − 1 × (36/37) = −1/37 ≈ −0.027

On average you lose about 2.7% of everything you stake. No sequence of bets — doubling after losses, switching colors, waiting for "cold" numbers — can turn a series of negative-expectation bets into a positive one. A system can change how often you win small amounts and how badly you occasionally lose, but not the average.

Treat gambling as paid entertainment with a predictable cost, never as income. Set a spending and time limit before you start. If gambling stops feeling like a choice, contact a local problem-gambling support service or a healthcare professional.

Three ways to read a probability

Comparison of common interpretations of probability (scroll sideways on small screens)
Interpretation	What "probability" means	Example	Works well when	Watch out for
Classical	Favorable ÷ possible equally likely outcomes	P(sum of 7) = 6/36	Symmetric devices: dice, cards, coins	Assuming symmetry that isn't there
Frequency	Long-run proportion in repeated trials	Share of defective parts in production	The process can be repeated under stable conditions	One-off events with no meaningful repetition
Bayesian (degree of belief)	Coherent degree of belief, updated with evidence	Belief a new drug works, revised after a trial	Combining prior knowledge with new data	Priors chosen to reach a preferred answer

What a probability can justify

Comparing options, pricing risk, planning for a range of outcomes and recognizing when a "pattern" is ordinary chance.

What it cannot justify

Certainty about any single event. A 90% forecast that turns out wrong was not necessarily a bad forecast — judge forecasts across many cases.

Sources and next steps

Blitzstein & Hwang, Introduction to Probability; Feller, An Introduction to Probability Theory and Its Applications; Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung (1933); OpenIntro Statistics, chapter 3. Next, learn to summarize what actually happened in Describe What You Observe.

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