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How to Draw a Tree Diagram (Mathematics Lesson)

What Is a Tree Diagram?

A tree diagram displays the possible outcomes from an event, and allows the probabilities of different outcomes to be calculated.

How to Draw a Tree Diagram

Question: Draw a tree diagram for the tossing of a coin:

Step 1: List the possible outcomes of the event.
The possible outcomes of tossing a coin are the coin coming up Heads or Tails.



Step 2: Draw a branch for each outcome of the event, being sure to label it.



Step 3: For each outcome, find the probability, and write it by the relevant branch.
The probability of getting a Head is ½, and the probability of getting a Tail is also ½.


This is the tree diagram that represents the tossing of a coin.

A Real Example of How to Draw a Tree Diagram

The above example showed a tree diagram for a single event.

It is possible to draw a tree diagram for several, sequential events.

For example, imagine tossing a coin twice, one time after the other.

The 1st toss is represented by the same tree diagram as above.



The 2nd toss is drawn in a similar way. But it is drawn to the right of the 1st toss. It must also be drawn for each of the outcomes of the previous toss:









This is the tree diagram that represents two tosses of a coin.

Another Real Example of How to Draw a Tree Diagram

The above example was relatively simple because the tossing of a coin is an independent event. The probability of getting a Head or Tail is always the same.

Drawing a tree diagram for a dependent event is more complicated.

The slider below gives a real example of how to draw a tree diagram for sequential dependent events.
Interactive Test
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Note
WHAT IS PROBABILITY?

Probability tells us how likely something is to happen.

The probability of an event is defined as:



Probability is given as a number between 0 and 1.

A probability of 0 means an event is impossible.

A probability of 1 means an event is certain.

The closer a probability is to 0 the less likely it is. The closer a probability is to 1 the more likely it is.



SIZE OF TREE DIAGRAMS

Tree diagrams can be any size.

It would be possible to draw a tree diagram for three coin tosses:



Top Tip

A USEFUL CHECK

There is a useful way to check that the probabilities on a tree diagram are all right.

Notice that at each branch, the probabilities add up to 1:



Read more about adding fractions