Probability from Outcomes

Statement

The probability of an event is given by:

\[ P(A) = \frac{\text{number of favourable outcomes}}{\text{total number of equally likely outcomes}} \]

Why it’s true

• Probability is a measure of how likely an event is.
• If all outcomes are equally likely, the chance of any specific event is the fraction of outcomes in its favour compared to the total outcomes.
• The probability always lies between 0 and 1.

Recipe (how to use it)

1. List all possible outcomes (denominator).
2. Count how many of these outcomes match event A (numerator).
3. Form the fraction and simplify if needed.

Spotting it

This formula is used in dice, coins, cards, balls from a bag, or any situation where outcomes are equally likely.

Common pairings

• Complement rule (\(P(A')=1-P(A)\)).
• Venn diagrams and probability addition rule.
• Experiments with dice, spinners, or card decks.

Mini examples

1. Coin flip: Favourable=1 (Heads), Total=2 → \(P(\text{Heads})=1/2\).
2. Die roll: Favourable=3 (2,4,6), Total=6 → \(P(\text{Even})=3/6=1/2\).

Pitfalls

• Not all outcomes are equally likely (then formula doesn’t apply directly).
• Forgetting to count outcomes correctly (e.g., suits in a deck of cards).

Exam strategy

• Draw the sample space if confused.
• Always check denominator = total outcomes.
• Reduce fractions to simplest form.

Summary

The probability of an event = favourable outcomes ÷ total outcomes. It is the foundation of probability theory.