Combined Events & Tree Diagrams (Years 8–11) — Maths

Flip a coin twice, or roll two dice, or draw two counters from a bag — very often the question isn't about one event but about two happening together. The whole topic turns on two tiny words: AND and OR. Spot which one the question is really asking and you already know whether to multiply or add.

Why this matters: real life is rarely a single roll of the dice. Whether two trains are both on time, whether you pass two exams, whether a machine's two parts both work — these are combined events. Getting AND/OR right is the foundation for everything from board games to insurance to conditional probability later.

A tree diagram: multiply probabilities along the branches

The AND / OR rules

AND narrows down (smaller); OR gathers up (bigger).

Everything in this chapter grows from one simple pair of rules:

AND (both events happen) → multiply the probabilities.
OR (either event happens) → add the probabilities.

Why multiply for AND? Think of it as narrowing down. Getting a head on the first coin already cuts your possibilities in half; needing a head on the second cuts those in half again. Half of a half is a quarter — and 1/2 × 1/2 = 1/4. Multiplying makes the answer smaller, which is right, because demanding two things at once is harder than demanding one.

Two coins, P(two heads): head AND head → 1/2 × 1/2 = 1/4. And indeed only 1 of the 4 equally-likely outcomes (HH, HT, TH, TT) is two heads.

Why add for OR? "OR" means you'll be happy with several different outcomes, so you gather them up. If a spinner gives red with probability 1/4 or blue with probability 1/4, then red OR blue = 1/4 + 1/4 = 1/2. (This clean adding works when the outcomes can't both happen at once.)

Question word	What it wants	Do this	Effect on answer
AND / both	every event together	multiply	gets smaller
OR / either	any one of them	add	gets bigger

Tree diagrams — a map of every possibility

Multiply across each path; the four paths add down to 1.

A tree diagram is like a signpost at every fork in the road: each branch is a choice, and following a full path from left to right traces one complete outcome. Two rules keep you right:

The branches leaving any single point always add up to 1 — something must happen.
To find the probability of a particular path, multiply along its branches (that's the AND rule in action). To combine several paths that all count as a "win", add down your results (that's the OR rule).

Two coins on a tree: the path H then H has probability 1/2 × 1/2 = 1/4. The four paths (HH, HT, TH, TT) each come to 1/4, and they add to 1 — a handy check that you've drawn it correctly.

The "at least one" trick

Go through the back door: find the small none slice, subtract from 1.

"At least one" hides a lot of paths (one, or two, or three...). Chasing them all is slow and error-prone. The clever shortcut is to go through the back door: the opposite of "at least one" is "none". Find P(none) — usually just one easy multiplication — then subtract from 1.

P(at least one) = 1 − P(none)

P(at least one head) on two coins: P(no heads) = 1/2 × 1/2 = 1/4, so P(at least one) = 1 − 1/4 = 3/4.
P(at least one six) on two dice: P(no six) = 5/6 × 5/6 = 25/36, so P(at least one six) = 1 − 25/36 = 11/36.

Common mistake: don't add when the question means AND. "A head and a six" is not 1/2 + 1/6 — that's bigger than either event alone, which is nonsense for "both". It's 1/2 × 1/6 = 1/12. Match the word to the operation before you touch the numbers.

In a nutshell: AND → multiply, OR → add. On tree diagrams, multiply across and add down; and for any "at least one" question, do 1 − P(none).