GCSE

Probability Trees

Diagrams that map out every possible outcome — and the two rules that make them work.

What Your Child Is Learning

A probability tree is a diagram that shows sequences of events. Each "branch" represents one possible outcome, and the number on each branch is the probability of that outcome happening.

Trees are used when events happen one after another — flipping a coin twice, picking two sweets from a bag, or checking whether it rains on two consecutive days. They make it possible to calculate combined probabilities without getting lost in the logic. With sweets taken out of a bag and not replaced, the second-branch probabilities change.

The two golden rules:
Along a branch (AND) = multiply.
Between branches (OR) = add.

How Schools Teach It Now

The method is systematic:

Draw the tree: first event on the left, branches splitting for each outcome. Second event branches off each first-event outcome.
Label every branch with its probability. At each split, the probabilities must add up to 1.
Multiply along a path to get the probability of that specific combination.
Add paths together when the question asks for "at least one" or "either...or".

Key Check

When you add up all the end probabilities (every possible path through the tree), they must total exactly 1. If they don't, there's an error somewhere.

Common Mistakes to Watch For

Adding when they should multiply: going along a branch means AND, which means multiply. Students often add the probabilities along a path. If P(rain) = 0.3 and P(rain again) = 0.3, the probability of rain on both days is $0.3 \times 0.3 = 0.09$, not $0.3 + 0.3 = 0.6$.
Branches not summing to 1: if P(rain) = 0.3, then P(no rain) must be 0.7. If the branches at any split don't add to 1, every calculation downstream will be wrong.
Missing paths for "at least one" questions: "at least one head in two flips" means HH, HT, or TH — three paths, not one. Students often forget to include all the relevant combinations. The shortcut: $P(\text{at least one}) = 1 - P(\text{none})$.

Try It Together

Worked Example — Rain Over Two Days

Given: the probability of rain on any day is 0.3, assuming the two days are independent (the question will tell you to).

Question 1: What is the probability it rains on both days?

Follow the "rain" branch twice — multiply along the path:

$$P(\text{rain both days}) = 0.3 \times 0.3 = 0.09$$

Question 2: What is the probability it rains on at least one of the two days?

Use the shortcut: find the probability of no rain on either day, then subtract from 1:

$$P(\text{no rain either day}) = 0.7 \times 0.7 = 0.49$$ $$P(\text{rain at least once}) = 1 - 0.49 = 0.51$$

Check: the chance of rain at least once must be more than 0.3 (one day alone gives that) and less than 0.6 (adding double-counts the both-days case). 0.51 fits.

Let Them Practise

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