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Class 12 Statistics Notes · GSEB

Mathematical Definition

Probability — learn the classical definition P(A) = m/n, its assumptions, key results, and limitations with a die calculator. GSEB Class 12 Statistics notes.

Last updated: 25 Aug 2026

Notes

Mathematical Definition of Probability

The classical (mathematical) definition assumes all outcomes in the sample space are equally likely. Probability is the ratio of favourable outcomes to total outcomes.

The Classical Definition

Classical Probability
If a random experiment has n equally likely outcomes and event A contains m of them, then P(A) = m/n. This definition requires two conditions: (1) The sample space is finite. (2) All outcomes are equally likely.

Classical Definition

P(A)=mnP(A) = \frac{m}{n}

Key Properties

0 ≤ P(A) ≤ 1
Probability is always between 0 and 1
P(S) = 1
Certain event has probability 1
P(∅) = 0
Impossible event has probability 0

Try It: Die Probability Calculator

Click on die faces to select favourable outcomes. Watch P(A) update live. Try the preset events or create your own.

Die Probability Calculator

Click on die faces to select outcomes, or use a preset above.

Event: No event selected

Click at least one face to compute P(A).

Solved Examples

Solved Example

Problem

A die is rolled. Find P(getting a number greater than 4).

Solution

P(number > 4) = 1/3 ≈ 0.333

Solved Example

Problem

Two coins are tossed. Find P(at least one head).

Solution

P(at least one head) = 3/4 = 0.75

Solved Example

Problem

A card is drawn from a 52-card deck. Find P(getting a king).

Solution

P(king) = 1/13 ≈ 0.077

Important Results

P(A) + P(A') = 1

An event and its complement always sum to 1.

For an elementary event: P = 1/n

A single outcome in equally likely sample space.

P(A) = 1 − P(A')

Useful when A' is easier to compute than A.

If A ⊂ B, then P(A) ≤ P(B)

Smaller event → smaller (or equal) probability.

Limitations of the Classical Definition

When classical definition fails

If outcomes are NOT equally likely, the formula P = m/n gives wrong answers. Example: A biased coin has P(H) ≠ 1/2. For biased experiments, we need the statistical (frequency-based) definition.

Key Takeaways

Key Takeaways

  • P(A) = m/n — favourable outcomes divided by total outcomes.
  • Works only when all outcomes are equally likely and the sample space is finite.
  • 0 ≤ P(A) ≤ 1 always. P(S) = 1 (certain), P(∅) = 0 (impossible).
  • P(A) + P(A') = 1 — use this when computing P(A) is hard but P(A') is easy.
  • For biased experiments (unfair coin, loaded die), classical definition does NOT apply.

Practice

  1. A die is thrown. Find P: (a) a prime number, (b) a number divisible by 3, (c) neither a prime nor a divisible by 3.
  2. From a bag of 6 white and 4 black balls, one ball is drawn at random. Find P(it is black).
  3. Two dice are thrown. Find P(sum is 8).

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