🎲 Probability

Probability#

A number in [0, 1] quantifying how likely an event is.

Important

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What it is#

Probability is a number between 0 and 1 that expresses how likely an event is: 0 means impossible, 1 means certain, and values in between are degrees of likelihood. A fair coin landing heads has probability 0.5.

The basic formula#

When every outcome is equally likely, probability is the ratio of favourable to total outcomes:

\[P(E) = \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}.\]

Rolling a 4 on a fair die: \(P = 1/6 \approx 0.167\).

Three ways to assign it#

Theoretical — from the structure of the problem (a fair die, a fair coin).
Experimental (frequentist) — from observed frequencies (flip a coin 100 times and count heads); it converges to the theoretical value as trials grow.
Subjective (Bayesian) — a degree of belief (a forecaster’s “70% chance of rain”).

The core rules#

Complement — \(P(\text{not } A) = 1 - P(A)\).
Addition (OR) — in general \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\), which reduces to \(P(A) + P(B)\) only when the events are mutually exclusive.
Multiplication (AND) — in general \(P(A \cap B) = P(A)\,P(B \mid A)\), which reduces to \(P(A)\,P(B)\) only when the events are independent. Rolling a 2 then a 5 on two dice: \(\tfrac{1}{6} \times \tfrac{1}{6} = \tfrac{1}{36}\).

Pitfalls and edge cases#

Mutually exclusive is not independent — exclusive events cannot co-occur (and are strongly dependent); independent events have no influence on each other. Confusing the two is the classic probability error.
Conditional probability — \(P(B \mid A) = P(A \cap B)/P(A)\) underlies the general AND rule and, ultimately, Bayes’ theorem.
Normalisation — probabilities over all mutually exclusive outcomes must sum to exactly 1.

Theme: Probability & Statistics Foundations · All terminology

Hint

Mind map — connected ideas

Bootstrap Confidence Intervals (CIs) · Mann–Whitney U Test (also called the Wilcoxon rank-sum test)

Hint

More in Probability & Statistics Foundations

Beta Distribution · Confidence Level · Correlation · Critical Value · Cumulative Distribution Function (CDF) · Frequentist · IID (Independent and Identically Distributed) · Likelihood · Margin of Error (MoE) · Mean · Median · Normal Distribution · Outlier · Population Proportion