Tree Diagrams
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- Tree Diagrams - Video - Multi step experiments using tree diagrams Watch
Theory
A tree diagram shows every outcome of a two-step experiment, with a probability on each branch. This Year 9 maths page shows how to multiply along a path, add the paths you need, change the second-stage branches when there is no replacement, use the complement for "at least one", and handle second stages whose probabilities depend on what happened first.
A tree diagram shows a multistage experiment. Each stage is a set of branches, labelled with the outcome and its probability.
A path runs from the start to one end branch, such as red then blue. Each end branch is one outcome of the whole experiment.
The branches leaving any one point add to \(1\). The probabilities of all the end branches also add to \(1\).
With replacement the second stage looks the same as the first (independent). Without replacement the second-stage probabilities change, because an item has gone (dependent).
A branch may only appear after some outcomes. If a second try only happens when the first try fails, the second stage is drawn on the "fail" branch only.
The two rules for trees:
How to use a tree diagram
- Draw the first stage: one branch per outcome, with its probability.
- Draw the second stage from each end. Change the fractions if there is no replacement, or if the question gives different probabilities after each first result.
- Check that each set of branches from a point adds to \(1\).
- Multiply along each path you need.
- Add those paths. For "at least one", find \(1 - P(\text{none})\) instead.
(i) \(P(\text{both yellow})\)
(ii) \(P(\text{different colours})\).
With replacement, both draws have \(P(\text{G}) = \dfrac{3}{8}\) and \(P(\text{Y}) = \dfrac{5}{8}\).
(i) Multiply along the path YY.
| \(P(\text{YY})\) | \(=\) | \(\dfrac{5}{8} \times \dfrac{5}{8}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{25}{64}}\) |
(ii) Different colours means the path GY or the path YG. Add the two paths.
| \(P\) | \(=\) | \(\dfrac{3}{8} \times \dfrac{5}{8} + \dfrac{5}{8} \times \dfrac{3}{8}\) |
| \(\) | \(=\) | \(\dfrac{30}{64} = \boldsymbol{\dfrac{15}{32}}\) |
(i) \(P(\text{same colour})\)
(ii) \(P(\text{at least one red})\).
After the first apple, \(6\) are left, with one fewer of the colour taken.
| \(P(\text{GG})\) | \(=\) | \(\dfrac{4}{7} \times \dfrac{3}{6} = \dfrac{12}{42}\) |
| \(P(\text{RR})\) | \(=\) | \(\dfrac{3}{7} \times \dfrac{2}{6} = \dfrac{6}{42}\) |
(i) Same colour is GG or RR.
| \(P(\text{same})\) | \(=\) | \(\dfrac{12}{42} + \dfrac{6}{42}\) |
| \(\) | \(=\) | \(\dfrac{18}{42} = \boldsymbol{\dfrac{3}{7}}\) |
(ii) The complement of "at least one red" is GG.
| \(P\) | \(=\) | \(1 - \dfrac{12}{42}\) |
| \(\) | \(=\) | \(\dfrac{30}{42} = \boldsymbol{\dfrac{5}{7}}\) |
(i) both spins are the same colour
(ii) the two spins are different colours.
(i) Same colour is three paths: RR, BB and GG. Multiply along each one, then add.
| \(P(\text{same})\) | \(=\) | \(0.5^2 + 0.3^2 + 0.2^2\) |
| \(\) | \(=\) | \(0.25 + 0.09 + 0.04\) |
| \(\) | \(=\) | \(\boldsymbol{0.38}\) |
(ii) Different colours is the complement of the same colour.
| \(P(\text{different})\) | \(=\) | \(1 - 0.38\) |
| \(\) | \(=\) | \(\boldsymbol{0.62}\) |
Check: the nine end probabilities add to \(1\).
The bus can be late in two ways. Multiply along each path, then add.
| \(P(\text{late})\) | \(=\) | \(0.25 \times 0.4 + 0.75 \times 0.1\) |
| \(\) | \(=\) | \(0.1 + 0.075\) |
| \(\) | \(=\) | \(\boldsymbol{0.175}\) |
Common pitfalls
Frequently asked questions
How do you use a tree diagram in probability?
Draw a set of branches for each stage, labelled with probabilities. Multiply along each path to get its probability, then add the paths that make up the event.
When do you multiply and when do you add on a tree diagram?
Multiply along a path (this and then that). Add different paths (this path or that path).
How does a tree diagram change without replacement?
On the second stage, the total goes down by one, and the count of the item already taken goes down by one. So the second-stage fractions are different on each branch.
How do you check a tree diagram?
The branches from each point must add to \(1\), and the probabilities of all the end branches must add to \(1\).
How many end branches does a tree have?
Multiply the number of branches at each stage. A spinner with \(3\) colours spun twice gives \(3 \times 3 = 9\) end branches.