# Tree diagrams

**Tree diagrams**

Tree diagrams are used in Probability when representing a sequence of events. They are very useful because they can be used to show all the outcomes in a clear and concise way.

For example, using a pack of cards in this scenario, if there is 15 and 9 are black with 6 red, what is the decimal choosing a red card as random?

P(red card) = = 0.4

In the next round, the another card is selected and then replaced back in the pack. What is the probability of picked another red / black card after that? The probability tree below shows the possible outcomes:

**Probability without replacement**

In this scenario, the probability changes because once a selection is made, it is not returned back into the pool of possibilities therefore changing the probability in the second round from the first. This is also an example of a **dependent** **event**.

For example, there are 20 balls in a bag, 6 are green and 14 are blue. Once a ball is picked, it is not returned back to the bag. What is the

All of the possible outcomes can be shown using a probability tree diagram:

**Probability with more than two outcomes**

In a room of of 30 teachers, 24 have mobile phones, 18 have internet access at home and 14 have both. This can be represented using the following diagram:

What is the probability of selecting a teacher at random who has:

- Only a mobile phone?
- A mobile phone and/or internet

**Summary**

- The word “or” means that probabilities need to be added – this means that events are
**mutually exclusive**. - The word “and” means that that probabilities need to be multiplies together – this means that two events are
**independent.**