Tree diagrams are one way of calculating the Probability of multiple events. These events can be independent or dependent, and on the diagram, we need to be able to list all possible outcomes of each event. For example, we could have it raining on a Monday or not raining on a Monday. Another possibility would be flipping a coin and getting either heads or tails.
Example of a tree diagram
The diagram gets its name from the branches, which show the possibilities of each event. A tree diagram example is shown below. This shows the possibilities when we flip a fair coin twice. We denote heads by H and tails by T.
Example tree diagram showing the possibilities when flipping a fair coin twice, Tom Maloy, StudySmarter Originals
How do we draw tree diagrams?
To draw a tree diagram, we can follow a set method:
Step 1: Look at the first event, and see how many distinct possibilities could occur. We will then draw that many lines at a constant degree of separation.
Step 2: Label each possibility at the end of the line. It usually helps to abbreviate each option to save space, e.g. H=heads.
Step 3: Label each branch with a Probability, ensuring the Probability is in either decimal or fraction form.
Step 4: Repeat steps 1-3 for as many events as there are, beginning from the end of each branch every time.
Let's take a football tournament, with play extending so that the only two possibilities are win or lose. In the first match, a team has a 60% chance of winning. If they win in the first match, the chances of winning the second game extend to 80%, whereas if they lose, it decreases to a 40% chance of winning. Show this information in a tree diagram.
First, we will denote a win by W and a loss by L. The first event is the first match.
Step 1: There are two events, so we need to draw two lines.
Step 2: We will denote one of these lines with a W at the end and the other with an L. This looks like the below.
Depiction of a tree diagram event - Tom Maloy, StudySmarter Originals
Step 3: If there is a 60% chance of winning, this means there is a 40% chance of losing, as the two options must sum to 100%. In terms of decimals, this means we have a 0.6 chance of winning and a 0.4 chance of losing. We can now add this to the diagram. (If you need a refresher on this, revise your knowledge on converting decimals and percentages)
Tree diagram event with probabilities - Tom Maloy, StudySmarter Originals
Step 4: We now need to repeat this process for the next branches. As there are again two outcomes in the second event, we draw two branches off of each branch, and then we label these W and L to represent winning and losing.
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The probability of winning after already winning is 0.8, so the probability of losing after a win is 0.2. The probability of winning after a loss is 0.4, so the probability of losing the second match in a row is 0.6. We can now fill these probabilities in on our tree diagram.
Chain of tree diagram events with probabilities - Tom Maloy, StudySmarter Originals
How are tree diagrams used to find the probability?
To find the probability of a certain set of outcomes occurring, we multiply across the branches that represent the outcomes, and if needed, add the probabilities of these long branches.
Following on from the above example, find the probability of a team winning one match and losing another, in any order.
The first thing we are going to do is multiply along each branch, to get the probability of each outcome occurring. The results of this are below.
Tree diagram example - Tom Maloy, StudySmarter Originals
If we want one win and one loss, then the team can lose the first game and win the second, or win the first and lose the second. That means we need to add together P(W, L) and P(L, W), which gets us 0.12+0.16=0.28.
Example problems involving tree diagrams
Example 1:
I have ten balls in a bag; five are green, three are yellow, and two are blue. I take one ball out of the bag and do not replace it. I then take another ball.
Draw a tree diagram to represent this scenario
Find the probability of taking two balls of different colours.
What is the probability of choosing two balls, neither of which are yellow?
a) Let us first find the probability of each ball in the first ball draw. For green, we have , for yellow we have , and for blue, we have . We can display this information on a tree diagram, where we use B to represent blue, Y for yellow and G for green.
Tree diagram for question a) - Tom Maloy, StudySmarter Originals
When we have taken one green ball out, we have nine balls total left, with four green, three yellow and two blue, so the probability of choosing green is , choosing yellow has probability and blue has probability .
When we have taken one yellow ball out, we have nine balls total left, with five green, two yellow and two blue, so the probability of choosing green is , choosing yellow has probability and blue has probability .
When we have taken one blue ball out, we have nine balls total left, with five green, three yellow and one blue, so the probability of choosing green is , choosing yellow has probability and blue has probability . This is shown in the tree diagram below.
Tree diagram for question a) - Tom Maloy, StudySmarter Originals
We will now multiply through the branches to get the probabilities of each possibility.
Tree diagram for question a) - Tom Maloy, StudySmarter Originals
b) For two balls of different colours, we need to add the various branches. This gives us
c) For two balls, neither yellow, we again add branches. We get
Example 2:
Below is a tree diagram. Fill in the gaps, and then use it to find the probability of two R and one B and the probability of getting the same letter three times.
Tree diagram for question b) -Tom Maloy, StudySmarter Originals
In each pair of corresponding branches, the probability must sum to one. Where there is 0.7 in one branch, the corresponding branch must be marked by 0.3. The same goes for 0.4 with 0.6, 0.2 with 0.8 and 0.1 with 0.9. Filling these in, we get the result below. Once we have done this, we can multiply along each branch to show the probability of that branch. This is also shown on the diagram below.
Tree diagram for question b) -Tom Maloy, StudySmarter Originals
To get two R's and a B, we can go RRB, RBR or BRR, so we need to sum these probabilities together. P(RRB)+P(RBR)+P(BRR)=0.224+0.294+0.036=0.554
To get the same letter three times, we can either have BBB or RRR. Adding the probabilities results in 0.056+0.021=0.077.
Tree Diagram - Key takeaways
A tree diagram is a way of finding probabilities of successive events.
To find the probability of two events occurring, multiply along the branches of the probability tree of this occurring.
The probability of each branch is shown at the end.
It is of paramount importance to label branches clearly.
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