Meaning of Probability and Sample Space
Simple Explanation
The sample space (S) is the set of every possible outcome of an experiment. An event (E) is any subset of outcomes you care about. When every outcome is equally likely, the probability of an event is P(E) = n(E)/n(S) — the fraction of the sample space the event covers.
Why Do We Need It?
Probability gives a precise number (between 0 and 1) for how likely something is, replacing vague words like "likely" or "unlikely" with an exact, comparable value.
Formula
Probability of an Event
P(E) = n(E) / n(S)
The probability of an event equals the number of favourable outcomes divided by the total number of equally likely outcomes in the sample space.
- P(E)
- — the probability of event E, a number from 0 to 1
- n(E)
- — the number of outcomes that make up event E (favourable outcomes)
- n(S)
- — the total number of outcomes in the sample space S
When to use it: Whenever every outcome in the sample space is equally likely, and you need the probability of a specific event.
Worked Example
Find the probability of rolling an even number
A fair six-sided die is rolled. Find the probability of rolling an even number.
Why Does This Work?
When every outcome is equally likely, each outcome contributes an equal "share" of the total probability — so the fraction of outcomes belonging to the event is exactly the fraction of the total probability the event represents.
Real-Life Example
Weather forecasts
A weather forecast reporting "a 30% chance of rain" is stating a probability.
That percentage comes from analyzing many past days with similar conditions — the fraction of those days ("the sample space") that actually saw rain ("the event").
Practice
A bag has 4 red balls and 6 blue balls. What is the probability of drawing a red ball? (Give as a decimal.)
EasyCommon mistake
Using the number of favourable outcomes as the denominator instead of the total sample space size — the probability formula always divides by the TOTAL number of outcomes, not the number of favourable ones.
Quick Review
- P(E) = n(E) / n(S), when all outcomes are equally likely.
- The sample space S is every possible outcome; an event E is a chosen subset of them.
- Probability is always a number between 0 (impossible) and 1 (certain).