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
"Fair" means every outcome has an equal chance (like a proper coin or dice). "Biased" means the outcomes are equally likely (like a weighted dice).
Calculating theoretical probability
"Theoretical" just means you can work it out with pure logic and counting — you don't need to actually
run the experiment. There are two versions of the same idea, depending on what you're being asked.
Examiner Tip
Unless the question tells you exactly how to write your answer, leave probabilities as fractions —
simplified where possible. It's the safest, most exact format and avoids rounding errors.
Practice Question 1
A bag contains 12 counters: 5 are red, 4 are green, and 3 are yellow. One counter is picked at random. Find P(green).
Practice Question 2
A fair six-sided dice is rolled once. Let A = "the dice lands on a prime number." Write out n(A) and find P(A).
Finding missing probabilities
Here's one of the most useful facts in this entire chapter: every probability in a full sample space adds up to exactly 1.
This makes sense if you think about it — something from that list happen (that's what "certain" means), so all
the chances of every possible thing happening must sum to "certain," which is 1.
This means if you're given a table of probabilities with one missing, you can always find it by subtracting the rest from 1.
The complement of an event
The complement of event A is simply "A does NOT happen" — sometimes written A′ or "not A."
Because P(A happens) + P(A doesn't happen) must add to 1 (one of the two occur), we get a really handy shortcut:
Mutually exclusive events
Two events are mutually exclusive if they literally cannot both happen on the same trial.
For example, when you roll one dice, you can't land on "a 6" "an odd number" at the same time —
those two events are mutually exclusive.
Think of it like doors: if event A and event B are mutually exclusive, they're like two separate
doors — you can only walk through one at a time. Because they never overlap, you can just add the
probability of each door to get the chance of walking through "either one."
Watch out
The OR rule (adding probabilities) only works for mutually exclusive events. If two events
happen together — like "rolling an even number" and "rolling a number greater than 3"
(which overlap at 4 and 6) — simply adding will overcount and give the wrong answer. This chapter
only covers mutually exclusive cases, but keep this distinction in mind for later topics.
Practice Question 3 — Worked-style (based on Emilia's spinner)
A spinner has 5 outcomes: Blue, Yellow, Green, Red, Purple.
P(Yellow) = 0.2, P(Green) = 0.1, P(Purple) = 0.4, and the spinner has an equal chance of landing on Blue or Red.
(a) Find P(Blue) and P(Red).
(b) Find P(Green or Purple).
(c) Find P(not Yellow).
2. Relative & Expected Frequency
What is relative frequency?
So far, everything has been theoretical — worked out with pure logic before anything even happens.
But sometimes you work out a probability theoretically. What's the probability a randomly
chosen person is left-handed? You can't reason your way to that with counting outcomes — you need real data.
That's where relative frequency comes in. It's an estimate of probability, built from actually
running the experiment and counting what happened.
Think of it like tasting soup: you can't know the exact seasoning of an entire pot just by looking,
so you take a spoonful (a sample of trials) and taste it. The more spoonfuls you try, the more confident
you become about the whole pot's flavour. Similarly, relative frequency gets more accurate the more trials you run.
Crucial distinction
Relative frequency is an estimate, not the true probability. Theoretical probability
(calculated by counting outcomes) is exact and doesn't need any data. Relative frequency
(calculated from experimental results) is our , and it only gets closer to
the true value as the number of trials increases.
Using relative frequency to test fairness
One of the most powerful uses of relative frequency is checking whether something is fair (unbiased)
or biased. You compare the experimental relative frequency to the known theoretical probability:
- Flip a coin 100 times, get 48 heads → relative frequency = 0.48. This is very close to the theoretical 0.5, so the coin is probably fair.
- Flip a coin 100 times, get 13 heads → relative frequency = 0.13. This is far from 0.5, so the coin is probably biased.
There's an important condition for relative frequency to be valid: each trial needs an equal chance
of success and trials must be independent of each other. If you're drawing objects from a bag,
that means you must replace the object each time — otherwise the probabilities shift as the bag empties,
and your "experiment" is no longer consistent from trial to trial. The experiment must also be genuinely
random, or you risk introducing bias into your results without realising it.
Practice Question 4 — Worked-style (based on Johan's buttons)
A bag contains an unknown number of different coloured buttons. Johan draws a button at random, notes its
colour, and replaces it. He repeats this 30 times and gets a red button on 18 occasions.
Estimate the probability that a button drawn from the bag is red.
Expected frequency
Once you know a probability (either theoretical or from relative frequency), you can flip the question
around: "if I repeat this experiment N times, how many successes should I expect?" That's expected frequency.
Think of it like a batting average: if a cricketer scores a boundary 1 time in every 4 balls
(probability = 0.25), and you know they'll face 40 balls today, you'd around 0.25 × 40 = 10
boundaries. It's a prediction based on the pattern so far — not a guarantee.
Examiner Tip
Exam questions won't always say "expected frequency" outright — they might phrase it as
Recognise this phrasing as your cue to multiply
probability × number of trials.
Practice Question 5 — Worked-style (bag of counters)
A bag contains 6 blue, 4 red, and 5 yellow counters. One counter is drawn at random, its colour noted,
then returned to the bag.
(a) Find the probability that a counter drawn is yellow.
(b) If this experiment is repeated 300 times, how many times would you expect a yellow counter to be drawn?
Practice Question 6
A biased coin is flipped 40 times and lands on heads 10 times. Based on this relative frequency,
how many heads would you expect if the same coin were flipped 100 times?
Exam Tips & Common Mistakes
Forgetting probabilities must sum to 1. Students often forget this rule exists and try to
guess a missing probability instead of subtracting the others from 1. This is one of the most
commonly tested ideas in this chapter — always check if a table's probabilities are "missing" one value first.
Treating 1 as prime. When listing prime numbers (e.g. for dice questions), remember 1 is
a prime number. The primes on a standard dice are only 2, 3, and 5.
Adding probabilities for events that aren't mutually exclusive. The OR rule (P(A) + P(B))
only works when the two events can never happen at the same time. Always check for overlap before adding.
Confusing relative frequency with theoretical probability. If a question gives you experimental
results (like "a coin was flipped 50 times and landed on heads 22 times"), you must use the relative
frequency formula — not just assume 0.5 because "it's a coin."
Not simplifying fractions. Mark schemes often expect probabilities in simplest form
(e.g. 1/3 instead of 5/15). Always check if your fraction can be reduced.
Forgetting to replace items when relative frequency requires independence. If a question
involves drawing from a bag multiple times to build a relative frequency, and it doesn't mention
replacement, double check — without replacement, each trial's probability changes, breaking the
assumption relative frequency relies on.
What examiners look for
Clear, labelled working. Write out the formula you're using (e.g. "P(A) = outcomes ÷ total") before
substituting numbers — this earns method marks even if your final answer has a small arithmetic slip.
For relative frequency and expected frequency questions, always state which numbers are the "successes"
and which are the "total trials" before dividing or multiplying.