Probability
Measuring chance without predicting it. A vocabulary (sample space, events), a handful of calculation rules, a tree that gets weighted and turned around, and a headline distribution, the binomial. The whole chapter rests on combinatorics: count the cases, then weigh them.
0The symbols
The chapter speaks the language of sets: every symbol translates a sentence about chance.
| Symbol | Meaning |
|---|---|
| Ω | The sample space: the set of all possible outcomes of the experiment. For a die, Ω = {1, 2, 3, 4, 5, 6}. |
| A, B | Events: subsets of Ω. "Rolling an even number" is the event {2, 4, 6}. |
| A̅ | The complement: everything not in A. It occurs exactly when A fails. |
| A ∪ B | "A or B": at least one of the two happens. The mathematical "or" is not exclusive. |
| A ∩ B | "A and B": both happen at once. If A ∩ B = ∅, the events are mutually exclusive. |
| P(A) | The probability of A: a number between 0 and 1 that measures the chance of A. P(Ω) = 1, P(∅) = 0. |
| P(A | B) | Conditional probability: the chance of A given that B has occurred. The sample space shrinks to B. |
| X | A random variable: a number attached to each outcome. The payoff of a game, the sum of two dice. |
| E(X) | The expected value: the average of the values of X, weighted by their probabilities. The average long-run payoff. |
| σ | The standard deviation: the typical spread of X around its expected value. Its square is the variance V(X). |
| B(n, p) | The binomial distribution: the number of successes in n independent repetitions of a trial that succeeds with probability p. |
1The language of events
Before calculating, you need to be able to say it. "Or", "and", "not", "at least": every word of the problem becomes a set operation.
1.1 · Sample space and events
A random experiment is an experiment whose possible outcomes are all known in advance, without being able to predict which one will occur. The sample space Ω gathers them all; an event is a subset of Ω, and it "occurs" when the outcome obtained belongs to it.
1.2 · The calculation rules
A and A̅ share the sample space without overlapping: their probabilities add up to 1. The complement rule is nothing more than that, but it is the most profitable tool in the chapter: "at least one" is almost always computed as "one minus none".
For the union, the diagram in section 1.1 supplies the proof: P(A) + P(B) counts the intersection twice, so we subtract it once. If A and B are mutually exclusive, the intersection is empty and the formula reduces to simple addition.
2Equally likely outcomes
When every outcome has the same chance, probability becomes pure counting: this is the bridge between this chapter and the previous one.
In a finite sample space where every outcome carries the same weight, the probability of an event is the proportion of outcomes it contains. Count the numerator, count the denominator: the four formulas of combinatorics become the two floors of a fraction.
The assumption does all the work: "fair", "at random", "well shuffled" are the words that, in a problem statement, license Laplace. Without them, the formula is illegal.
The classic trap is a sample space whose outcomes don't carry equal weight. The sum of two dice takes eleven values, from 2 to 12, but these eleven values are not equally likely: you have to go down to the 36 ordered pairs, which are.
3Frequency and chance
What does "six comes up with probability 1/6" mean for a die rolled only once? The experimental answer: roll it many times. The relative frequency of each face settles around its probability, that is the law of large numbers.
Try it here: roll 10 times, then 10,000. Watch the disorder of small samples, then the order that sets in. The ocre marks show the theoretical values.
4Conditional probability and the tree
Information changes the odds. "Given that" is the most important phrase in the chapter, and the weighted tree is its calculating machine.
4.1 · Given that
Learning that B has occurred shrinks the sample space: outcomes outside B are eliminated. The chance of A is recalculated within this reduced world, dividing by P(B) so the new total comes to 1.
4.2 · The weighted tree
The tree encodes three rules in one picture. Along a path, you multiply: that is the definition of conditional probability, read backwards. Branches from the same node total 1: the complement rule demands it. And the probability of a final event is the sum of every path leading to it: total probability.
Discipline pays off: every weight on the second tier is a conditional probability, computed in the world where the first tier has already happened.
5Turning the tree around: Bayes
The tree gives P(test positive | sick). The doctor's question runs the other way: P(sick | test positive). Bayes' theorem makes the U-turn, and its result surprises almost everyone.
Set the disease and the test, and read the answer in head counts: out of 10,000 people, the false positives from the large healthy population often swamp the true positives from the small sick population.
The formula holds no mystery: it is the definition of conditional probability, applied twice to the same P(A ∩ B). What is surprising is the weight of prevalence: an excellent test for a rare disease mostly produces false positives, because the healthy vastly outnumber the sick.
The reflex that saves you: translate into head counts. "Out of 10,000 people..." makes visible what percentages hide.
6Independence
Two events are independent when knowing that one has occurred changes nothing about the odds of the other. Then, and only then, "and" is computed by simple product.
This is the product rule stripped of its condition: if P(B | A) = P(B), then P(A ∩ B) = P(A) · P(B). Two rolls of a die, two draws with replacement: independent. Two draws without replacement: dependent, the urn remembers.
Do not confuse independent with mutually exclusive: mutually exclusive events with nonzero probabilities are violently dependent, since one rules out the other.
7Expected value
A random variable attaches a number to each outcome: a payoff, a sum, a count. Its expected value is the average of these numbers, each weighted by its probability: what the game pays on average, over the long run.
The law of large numbers from section 3 gives the meaning: the average of observed results converges to E(X). An expected value is therefore not a prediction for tomorrow, it is a balance sheet for a thousand tomorrows.
A game is fair when the expected net payoff is zero. Casino games never are: their expected value is negative for the player, that is the house's business model.
8The binomial distribution
Repeat a trial n times, independently, where it succeeds with probability p, and count the successes: that is the binomial distribution. It brings together this whole chapter and the previous one: independence gives pᵏ(1 − p)ⁿ⁻ᵏ, combinatorics gives the Cₙᵏ.
Explore the whole distribution: n and p draw the histogram of every P(X = k). Click a bar to read its exact value and the cumulative. The ocre mark under the axis shows the expected value np.
9The method
Five steps, in order. The first four choose the tool; the fifth avoids embarrassment.
10Six mistakes
11Test yourself
Eight questions, exactly one correct answer each time. The explanation appears after your choice.
12The vocabulary in French and German
This formulary comes from Swiss gymnasium (high school) classrooms; here are the terms in their original languages.
| Français | Deutsch |
|---|---|
| la probabilité | die Wahrscheinlichkeit |
| l'expérience aléatoire | das Zufallsexperiment |
| l'univers | der Ergebnisraum |
| l'issue | das Ergebnis |
| l'événement | das Ereignis |
| l'événement contraire | das Gegenereignis |
| incompatibles / disjoints | unvereinbar / disjunkt |
| équiprobable | gleichwahrscheinlich |
| cas favorables / possibles | günstige / mögliche Fälle |
| la fréquence relative | die relative Häufigkeit |
| la loi des grands nombres | das Gesetz der grossen Zahlen |
| la probabilité conditionnelle | die bedingte Wahrscheinlichkeit |
| sachant que | unter der Bedingung, dass |
| l'arbre pondéré | das gewichtete Baumdiagramm |
| les probabilités totales | die totale Wahrscheinlichkeit |
| la formule de Bayes | der Satz von Bayes |
| indépendant | unabhängig |
| avec / sans remise | mit / ohne Zurücklegen |
| la variable aléatoire | die Zufallsgrösse |
| l'espérance | der Erwartungswert |
| la variance / l'écart-type | die Varianz / die Standardabweichung |
| le jeu équitable | das faire Spiel |
| l'épreuve de Bernoulli | das Bernoulli-Experiment |
| le succès / l'échec | der Erfolg / der Misserfolg |
| la loi binomiale | die Binomialverteilung |
| le dépistage | das Screening / die Reihenuntersuchung |