Quick Summary
- Every probability is a number from 0 (impossible) to 1 (certain) — it can be written as a fraction, decimal, or percentage.
- An experiment is repeated in trials, and each trial produces an outcome. An event is one outcome or a group of outcomes you care about.
- Theoretical probability = (outcomes for the event) ÷ (total outcomes) — worked out with maths, no experiment needed.
- All probabilities in a sample space add up to exactly 1. This is how you find "missing" probabilities.
- The complement of an event ("not A") = 1 − P(A).
- Mutually exclusive events can't happen at the same time — you add their probabilities: P(A or B) = P(A) + P(B).
- Relative frequency is an estimate of probability from real experimental data: (successful outcomes) ÷ (total trials). More trials = more accurate estimate.
- Expected frequency predicts how many times something should happen: probability × number of trials.
1. Basic Probability
What is probability, really?
In everyday life, you already think in probability terms without realising it. You say things like
"it's to rain today" or "I'm I locked the door." Maths just takes those
fuzzy words and puts a precise number on them, using the probability scale — a number line from
0 to 1.
Think of it like a dimmer switch: 0 is the light completely off (this will never happen),
1 is the light at full brightness (this will definitely happen), and 0.5 is exactly halfway —
an even chance, like a coin flip. Every probability you'll ever calculate sits somewhere on that dimmer switch.
00.250.50.751
Impossible
Unlikely
Even chance
Likely
Certain
Probabilities can always be written three ways — and they all mean exactly the same thing:
| Fraction | Decimal | Percentage |
|---|
| 1/4 | 0.25 | 25% |
| 1/2 | 0.5 | 50% |
| 3/4 | 0.75 | 75% |
The key vocabulary
Before you can talk about probability properly, you need these words locked in — examiners use them precisely, and mixing them up costs marks.
Experiment
An activity you repeat to produce results — e.g. rolling a dice, flipping a coin, picking a marble from a bag.
Trial
One single repeat of the experiment. Roll the dice once = one trial.
Outcome
A possible result of one trial. Rolling a dice has 6 possible outcomes: 1, 2, 3, 4, 5, 6.
Event
An outcome, or a of outcomes, that you're interested in. "Rolling an even number" is an event made up of the outcomes {2, 4, 6}.
n(A)
The number of outcomes belonging to event A. If A = "dice lands even", then n(A) = 3, because {2, 4, 6} has 3 members.
Sample space
The complete set of of the experiment. For a dice, the sample space is {1, 2, 3, 4, 5, 6}.
P(A)
Notation meaning "the probability of event A happening."
Fair vs. biased
Calculating theoretical probability
2. Relative & Expected Frequency
What is relative frequency?
theoretical
relative frequency
estimate