Sheet № 122 · Foundation + Higher · AQA · Edexcel · OCR
Probability Tree Diagrams –
Tree diagrams are one of the most important tools in GCSE probability. They let you organise multi-stage events clearly, showing every possible outcome and its probability. Whether events are independent (with replacement) or dependent (without replacement), the method is the same: multiply along branches for AND, add between branches for
§Key definitions
Question:
A bag contains 4 red and 6 blue counters. A counter is picked at random, replaced, and a second counter is picked. Find the probability of getting two red counters.
Answer:
P(two red) = 4/25.
Q1 (Foundation):
A spinner has P(Win) = 0.4. It is spun twice. Find P(two wins).
Q2 (Foundation):
Using the same spinner, find P(exactly one win in two spins).
Q3 (Higher):
A box has 6 milk and 4 dark chocolates. Two are taken without replacement. Find P(both dark).
§Formulas to memorise
P(A and B) = P(A) × P(B) — multiply along branches
P(A or B) = P(A and B₁) + P(A and B₂) — add between outcomes
Draw the first set of branches — for the first event, labelling each outcome and its probability.
Draw the second set of branches — from each first-event outcome, updating probabilities if the events are dependent.
Multiply along branches — to find the probability of each combined outcome.
Add probabilities — of the outcomes that satisfy the condition in the question.
Worked example
A bag contains 4 red and 6 blue counters. A counter is picked at random, replaced, and a second counter is picked. Find the probability of getting two red counters.
Working: P(Red) = 4/10 = 2/5. P(Blue) = 6/10 = 3/5. Since the counter is replaced, probabilities stay the same. P(Red and Red) = 2/5 × 2/5 = 4/25.
⚠ Common mistakes
- ✗Adding instead of multiplying along branches. To find P(A and B), multiply the branch probabilities. Only add when combining separate outcomes at the end.
- ✗Not adjusting probabilities for without replacement. If a ball is not returned, the total and the count for that colour both change on the second pick. For example, after removing one green ball from 5 green and 3 yellow, there are 7 balls left.
- ✗Forgetting to consider both orders. "One of each" means Green-then-Yellow OR Yellow-then-Green. Both paths must be included.
- ✗Incorrect simplification of fractions. When multiplying fractions along branches, simplify only at the end. For example, 4/10 × 3/9 = 12/90 = 2/15, not 12/100.
✦ Exam tips
- →Always check that branches at each stage add up to 1. This is a quick error check.
- →Label your tree diagram clearly with outcomes and probabilities — examiners award marks for a well-drawn tree even if the final answer is wrong.
- →For "at least one" questions, it is often easier to calculate P(none) and subtract from 1: P(at least one) = 1 − P(none).
- →Leave probabilities as fractions throughout your working. Only convert to decimals at the end if the question requires it.
- →For without-replacement questions, write the adjusted totals clearly on the second set of branches to avoid errors.