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What Are Tree Diagrams in Maths?

A tree diagram is a diagram that presents multiple trials in a clear and easy-to-read manner. The branches of the tree represent the probabilities of the possible outcomes of each trial.

Theory

Tree Diagram

A tree diagram is a great way to structure different outcomes when you’re doing multiple trials. It’s also good at showing how many outcomes multiple trials has. If you have a large amount of trials, the tree diagram will become too large to give you a good overview.

Important things to remember about tree diagrams:

1.
When you move along a branch, you multiply the probabilities of each outcome for each trial with each other.
2.
When several branches fit your event, add the probabilities of those branches together.

In the tree diagram below, you can see that p1 and p2 are the probabilities of the two possible outcomes of each trial. To find the probabilities of the possible outcomes of the multiple trials, we have to combine these by multiplying the relevant ones together.

You can see from the tree diagram that when you have two trials that each has two possible outcomes, the possible outcomes of the multiple trials have the probabilities p1p1, p1p2, p2p1 and p2p2.

A tree diagram with two trials which have to outcomes

Example 1

Illustrate the different outcomes you can get by tossing a coin three times, using a tree diagram

There are two possible outcomes for each coin toss, head (H) and tail (T). Thus it becomes

2 2 2 = 8

possible outcomes when a coin is tossed three times. This is well illustrated by means of a tree diagram:

Tree diagram of probabilities of getting heads or tails by tossing a coin

Example 2

Tree Diagram

When giving birth, the probability of having a girl is 0.486, while the probability of having a boy is 0.514. The probability of the child being born colorblind is 0.044. Make a tree diagram to show the probabilities of having a girl or boy who is or isn’t colorblind.

From the text you can see that the trials are gender and colorblindness. That gives you the following possibilities:

  • The trial regarding the gender of the baby has only two possible outcomes: Boy or girl.

  • The trial regarding whether the baby is colorblind or not also has only two possible outcomes: Colorblind or not colorblind.

You already know the probabilities of both boy and girl, but you still only have the probability of the baby being colorblind. That means you have to find the probability of not-colorblind. Colorblind and not colorblind are complementary events, which gives you

P (not colorblind) + P (colorblind) = 1.

We find the probability: P (not colorblind) + 0.044 = 1 P (not colorblind) = 1 0.044 = 0.956

If you organize this in a tree diagram, it’s pretty apparent what you need to do, namely multiply the numbers along each branch with each other. The tree diagram ends up looking like this:

Tree diagram of probabilities of colorblindness for the different sexes

Here are the calculations:

P (not colorblind given boy) = 0.514 0.956 = 0.491 P (not colorblind given girl) = 0.486 0.956 = 0.465 P (colorblind given boy) = 0.514 0.044 = 0.023 P (colorblind given girl) = 0.486 0.044 = 0.021

P (not colorblind given boy) = 0.514 0.956 = 0.491 P (not colorblind given girl) = 0.486 0.956 = 0.465 P (colorblind given boy) = 0.514 0.044 = 0.023 P (colorblind given girl) = 0.486 0.044 = 0.021

This lets you read the probabilities of having a boy or a girl who is or isn’t colorblind straight off the tree diagram.

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