Statistical Definition of Probability — Class 12 Statistics Notes (GSEB)

Why a Second Definition?

The mathematical (classical) definition of probability requires the outcomes of the random experiment to be finite, known and equi-probable. When these conditions cannot be satisfied — because outcomes are infinite, unknown, or not equi-probable — the probability of an event cannot be obtained from the mathematical definition. In such cases the statistical definition is used. ⭐

Click each scenario — the test is: finite + known + equi-probable → classical; otherwise → statistical.

The Statistical Definition

Suppose the number of trials in a random experiment is n. If the event A occurs in m trials out of these n trials, then the ratio m/n is called the relative frequency of the event A in the n trials of the random experiment. If the number of trials of the random experiment is gradually increased infinitely, the relative frequency of the event A stabilizes near a fixed number. This fixed number is called the probability of the event A. The definition of the probability of an event as a limiting value of the relative frequency is called the statistical definition of probability.

Statistical definition ⭐

P(A) = \lim_{n \to \infty} \frac{m}{n}

where n = total number of trials (increased indefinitely) and m = number of trials in which event A occurs. ⭐

Relative Frequency Lab — watch m/n converge

The coin's true bias is hidden — until you reveal it

Hidden bias0.80

Run an experiment — the ratio of heads to total tosses will wobble at first, then settle.

The relative frequency stabilizes near a fixed number as n grows — that fixed number is the probability. Because we can never run infinitely many trials, the value is always an approximation “near” the limit, never exact.

Because we can never run infinitely many trials, statistical probabilities are always approximations settled “near” a fixed number — never exact. This is why the result of tossing a particular coin repeatedly is described as approaching a limit, not reaching one.

Limitations of the Statistical Definition

Two practical cracks in the definition. ⭐ Section C Q8

In this definition, the number of trials of the random experiment must be infinite to obtain the exact probability of the event, which is not practically possible.

The definition is based on the assumption that the relative frequency of the event remains more or less the same for a large number of trials in the random experiment, which is not always true in practice.

The fixed number to which the relative frequency tends “is called the probability of the event A” — Abraham de Moivre first suggested taking this limiting value as the definition of probability. (His famous 1718 book The Doctrine of Chances is the source being cited.)

Solved Illustrations 36–37

Solved Example

Problem

Illustration 36 (⭐ board — one-digit numbers 1 to 9, divisible by 3): the experiment is repeated with replacement n = 1000 times. Find the probability that the number is divisible by 3, by the statistical definition.

Solution

P(A) = 0.3 — the limiting value of the relative frequency.

Solved Example

Problem

Illustration 37: a hospital's maternity ward recorded 420 male births out of the last 1000 births. Find the probability of a male birth.

Solution

P = 0.42.

Key Takeaways

The statistical definition handles experiments the classical one cannot: infinite/unknown outcomes, or outcomes that are not equi-probable.
P(A) = lim n→∞ (m/n) — probability is the limiting value of the relative frequency, never the frequency of one particular run.
The definition is approximate in practice (trials cannot be infinite) and assumes the relative frequency stays stable for large n.
The classical and statistical definitions agree where both apply (the 1-to-9 digits: 0.3 observed ≈ 1/3 classical).
Board link: relative-frequency questions give you m and n directly and ask for P(A) = m/n; de Moivre's name is the book's historical footnote.