A probability tree diagram shows two or more events in stages, with the probabilities written on the branches. The branches from each point add up to 1. To find the probability of a combined outcome, multiply along its branches; if several outcomes satisfy the event, add their probabilities. For “at least one”, it is usually quickest to work out 1 − P(none).
Tree diagram rules
- Branches from one point add up to 1
- Multiply along a path for “and”
- Add the results of different paths for “or”
- P(at least one) = 1 − P(none)
- All final outcomes add up to 1
Drawing a tree diagram
Each stage of the experiment gets a set of branches, one for each possible outcome, labelled with its probability. The branches from any one point must add up to 1. For a biased coin with P(heads) = 0.65, each set of branches shows 0.65 for heads and 0.35 for tails.
Multiply along, add between
To find the probability of one particular path, such as heads then heads, multiply the probabilities along it: 0.65 × 0.65 = 0.4225. If an event can happen in more than one way, such as exactly one head, work out each path and add: 0.65 × 0.35 + 0.35 × 0.65 = 0.455.
The “at least one” shortcut
“At least one” covers many paths, so it is quicker to use the complement. The only outcome that is not “at least one tail” is heads, heads, so P(at least one tail) = 1 − 0.4225 = 0.5775. For three rolls of a die, P(at least one six) = 1 − (5/6)³ = 91/216.
When the second stage depends on the first
Sometimes the probabilities on the second set of branches depend on what happened first. If P(rain) = 0.3, P(late | rain) = 0.4 and P(late | no rain) = 0.1, the tree has different second-stage branches after rain and after no rain. P(late) = 0.3 × 0.4 + 0.7 × 0.1 = 0.19.
Checking your work
The probabilities of all the final outcomes of a tree should add up to 1. If they do not, check that each set of branches adds to 1 and that you have multiplied correctly.
Worked examples
Example 1. A biased coin has P(heads) = 0.65. It is flipped twice. Find P(two heads).
- Multiply along the heads–heads path: 0.65 × 0.65.
Answer: 0.4225.
Example 2. Using the same coin, find P(exactly one head).
- HT: 0.65 × 0.35 = 0.2275. TH: 0.35 × 0.65 = 0.2275.
- Add the two paths.
Answer: 0.455.
Example 3. Using the same coin, find P(at least one tail).
- P(none) is P(two heads) = 0.4225.
- 1 − 0.4225.
Answer: 0.5775.
Example 4. P(rain) = 0.3. If it rains, P(the bus is late) = 0.4; if not, P(late) = 0.1. Find P(the bus is late).
- Rain and late: 0.3 × 0.4 = 0.12.
- No rain and late: 0.7 × 0.1 = 0.07.
- Add.
Answer: 0.19.
Example 5. A fair die is rolled three times. Find P(at least one six).
- P(no sixes) = (5/6)³ = 125/216.
- 1 − 125/216.
Answer: 91/216.
Common mistakes
- Adding along a path. Multiply along a path; add between different paths.
- Branches that do not add up to 1. Each set of branches from one point must total 1.
- Missing a path for “exactly one”. Both orders count, such as HT and TH.
- Working out every path for “at least one”. Use 1 − P(none) instead.
- Using the same second-stage probabilities when they depend on the first. Check whether the second event depends on the first.
How tree diagrams are examined
Probability trees are Higher content on the UK GCSE and are used at both tiers of the International GCSE. Questions ask you to:
- complete the probabilities on a tree diagram
- find the probability of a combined outcome
- find the probability of “exactly one” or “at least one”
- handle a second stage that depends on the first
Use the tree diagrams worksheet, which includes answers, and the GCSE probability revision engine.
When items are not replaced, see without replacement. Trees rely on the AND and OR rules.
Frequently asked questions
How do you use a probability tree diagram?
Multiply along the branches for a single path, and add the results of different paths that satisfy the event.
What do the branches from one point add up to?
1.
How do you find P(at least one)?
Work out P(none) and subtract it from 1.