Hong Kong
Stage 1 - Stage 3

# Tree diagrams

Lesson

So far we have looked into 2 different types of visual representations for data relating to probability, the Venn Diagram and a Two Way Table.  We introduce a third type of representation now, the Tree Diagram.

A tree diagram is named because the diagram that results looks like a tree.

Sometimes calculating probabilities can be complicated, especially if the events have different weightings or are unequal events.

A tree diagram has 4 important components

• Seed
• Branch
• Probability (used when the events have unequal weightings)
• Outcome

.

### Trees for singular trials

When a single trial is carried out, we have just one column of branches.

Here are some examples.None of these have probabilities written on the branches because they all have equal chance of happening.

 The outcomes for tossing a coin once Outcomes from having a baby Outcomes from rolling a standard die

Here are some examples that have probabilities on the branches, because they have specific probabilities of happening.

 Showing outcomes of whether some friends received mail or not Showing outcomes of whether I won my tennis game or not Showing outcomes of whether I hit the dartboard or not

Things to notice about tree diagrams.

• When looking at a group of branches, the sum of the group should add to 1.  This indicates that all the outcomes are listed.

## Trees for multiple trials

When more than one trial is carried out, we have two (or more) columns of branches.

Here are some examples.  These ones without probabilities, because they have equal chance of happening.

 Tree diagram showing outcomes and sample space for tossing a coin twice to calculate the probability of HH we would have 1/4, or the probability of just 1 head, would be 2/4 as the outcomes HT and TH both have just one head. tree diagram showing outcomes and sample space for rolling a dice twice

Here is an example that have probabilities on the branches, because they have specific probabilities of happening. The probabilities of the events are multiplied through each branch.

Outcomes of playing two games of tennis.  Since I have a 3% chance of winning, the chance that I win 2 games in a row is 9%.

If I wanted to know what the chance is that I win at least 1, I could do either

a) add 9+21+21 together and get 51%,

or b) use the complementary event of losing both games and calculate 1-49 = 51%

Things to note from using probability trees to calculate probabilities of multiple trials.

• Multiply across the tree to calculate the probability of individual outcomes.
• Add down the list of outcomes to calculate the probability of multiple options.
• The final % should add to 100, or the final fractions should add to 1 - this is useful to see if you have calculated everything correctly.
Probability Relationships

P(A)+P(A') = 1

P(A U B) = P(A) + P(B) - P(AB)

Multiply through the branches of a tree to calculate probabilities of outcomes.

Add through the rows of the outcomes to calculate multiple options.

#### Worked Examples

##### QUESTION 1

Construct a tree diagram showing all the ways a captain and a vice-captain can be selected from Matt, Rebecca, Helen and Chris.

##### QUESTION 2

Construct a tree diagram showing all possible outcomes of boys and girls a couple with three children can possibly have.

##### QUESTION 3

A coin is tossed twice.

1. Construct a tree diagram showing the results of the given experiment.

2. Use the tree diagram to find the probability of getting exactly 1 Head.

3. Use the tree diagram to find the probability of getting 2 heads.

4. Use the tree diagram to find the probability of getting no heads.

5. Use the tree diagram to find the probability of getting 1 Head and 1 Tail.