Probability
Grade 5-7Tree Diagrams
A tree diagram lays out every possible combination of two or more events as branches, and two simple rules unlock it: multiply along a branch to find one specific outcome, and add branches together when more than one outcome counts. This lesson covers multiplying along branches, finding a missing probability, adding across branches for combined outcomes, and tree diagrams for picking without replacement.
What you need to know
- Each set of branches must have probabilities that add to 1.
- Multiply along the branches to get the probability of a whole path.
- Add the results of the different paths that satisfy the condition.
- Without replacement, the second set of probabilities changes because one item has been removed.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Multiplying along a branch
Each complete path through a tree diagram, from the start to one of the final outcomes, represents one specific combination of results. Multiply the probabilities along that single path to find the probability of that exact outcome.
A spinner with P(red) = 0.4 is spun twice. The probability of red then red is found by multiplying along that branch: 0.4 × 0.4 = 0.16.
Finding a missing probability
At any point where the branches split, the probabilities of all the branches from that point must add up to 1. Use this to find a missing probability by subtracting the known branches from 1.
A spinner has P(red) = 0.35, so P(blue) at the same point is 1 - 0.35 = 0.65.
Adding across branches for combined outcomes
When more than one path through the tree counts as a valid outcome, such as "exactly one" or "at least one", find the probability of each matching path separately by multiplying along it, then add these probabilities together.
A biased coin has P(heads) = 0.7 and is flipped twice. The probability of exactly one head comes from two paths, heads-then-tails and tails-then-heads: (0.7 × 0.3) + (0.3 × 0.7) = 0.21 + 0.21 = 0.42.
Tree diagrams without replacement
When an item is picked and not replaced, the total number of items left changes for the second pick, and so does the number of items matching each outcome if that outcome was picked first. The probabilities on the second set of branches depend on what happened on the first branch.
A bag has 5 red and 3 blue counters, 8 in total. After picking a red counter without replacing it, 4 red and 3 blue counters remain out of 7, so the second branch from "red" shows P(red) = 4/7 and P(blue) = 3/7.
Worked examples
Three exam-style questions, fully solved
A spinner can land on red or blue. P(red) = 0.35. The tree diagram shows the spinner spun twice. Find the missing probability of blue on the second spin, following a first spin of red.
Easy- 1.Recognise that the branches from the same point must add up to 1
- 2.Subtract the known probability from 1: 1 - 0.35
Answer: 0.65
A bag contains 5 red and 3 blue counters. Two counters are picked at random, without replacement. Use the tree diagram to find the probability that both are red.
Medium- 1.Find the probability of red on the first pick: 5/8
- 2.Find the probability of red on the second pick, given the first was red: 4/7, since one red counter has been removed
- 3.Multiply along the branch: 5/8 × 4/7
Answer: 5/14
A bag contains 4 red and 6 blue counters. Two counters are picked at random, without replacement. Use the tree diagram to find the probability of picking one red and one blue, in either order.
Hard- 1.Find the probability of red then blue: 4/10 × 6/9 = 24/90
- 2.Find the probability of blue then red: 6/10 × 4/9 = 24/90
- 3.Add the two matching branches together: 24/90 + 24/90
Answer: 8/15
Practice
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Practice
Now try these yourself.
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A bag has 3 red and 2 blue counters. One is taken, its colour noted, and it is replaced. A second is then taken. Work out the probability that both are red.
Using the same bag (3 red, 2 blue, taken with replacement), work out the probability that both counters are blue.
A bag has 4 red and 6 green counters. Two are taken without replacement. Work out the probability that both are red.
Using the bag of 4 red and 6 green (taken without replacement), work out the probability that both counters are green.
The probability that it rains on any day is 0.3, independently of other days. Work out the probability that it rains on two consecutive days.
On a tree diagram, one branch leaving a point has probability 0.35. What is the probability on the other branch from that same point?
Avoid these
The mistakes students make most often
Forgetting that the probabilities on branches from the same point must add up to 1 when finding a missing value.
Adding the probabilities along a single branch instead of multiplying them to find the probability of one specific path.
When a question needs "at least one" or "exactly one", only using one matching branch instead of adding together every branch that gives that outcome.
Using the same denominator for the second pick as the first pick when picking without replacement, instead of reducing it by one.
Forgetting that the numerator on the second branch also changes without replacement, depending on what was picked first.
FAQ
Questions students and parents ask
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