Probability and Tree Diagrams

GCSE Maths · Probability

Probability basics

Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). It can be written as a fraction, decimal or percentage.

P(event) = number of favourable outcomes ÷ total number of outcomes

Example: rolling a 3 on a fair dice → P(3) = 1/6.

The probabilities of all possible outcomes add up to 1.
So P(not A) = 1 − P(A) (the probability something does not happen).

Mutually exclusive events

Events that cannot happen at the same time (e.g. rolling a 2 or a 5). For these, add the probabilities: P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3.

Expected frequency

How many times you'd expect an event in a number of trials:

expected frequency = probability × number of trials

E.g. flipping a fair coin 80 times → expect 0.5 × 80 = 40 heads.

Experimental (relative frequency)

Based on actual results rather than theory:

relative frequency = number of times it happened ÷ total trials

The more trials, the closer this gets to the true (theoretical) probability.

Tree diagrams

Used for two or more events in sequence (e.g. two coin flips). Rules:

Multiply probabilities along the branches (AND).
Add the probabilities of the different end-results you want (OR).
Probabilities on each set of branches add up to 1.

Example: two flips of a fair coin. P(two heads) = ½ × ½ = ¼. P(exactly one head) = (½×½) + (½×½) = ½.

With and without replacement

With replacement: probabilities stay the same each time.
Without replacement: the total (and sometimes the favourable count) decreases for the second pick (e.g. taking counters from a bag).

Worked example

A bag has 4 red and 6 blue counters. One is picked at random. Find P(red), then P(not red).

P(red) = 4/10 = 2/5.
P(not red) = 1 − 2/5 = 3/5. ✓

Common mistakes

Adding probabilities along a tree branch instead of multiplying.
Forgetting to reduce the total for "without replacement" problems.
Probabilities that don't add up to 1 across all outcomes.

Exam tips

On tree diagrams: × along, + across the outcomes you want.
Use P(not A) = 1 − P(A) to save work.
Watch for "without replacement" — the second denominator changes.

Key facts to remember

P = favourable ÷ total; all outcomes sum to 1; P(not A) = 1 − P(A).
Mutually exclusive → add; tree diagrams → multiply along branches, add the desired outcomes.
Expected frequency = probability × trials; "without replacement" reduces the total each pick.