Tree Diagrams
Introduction
When studying GCSE Maths you will encounter a topic called probability and you may have met that topic already. An extension to probability is something called tree diagrams which can help us to find the probability when we are dealing with more than one event.
For instance you may be flipping a coin and rolling a dice and you may want to find the probability of the coin landing on tails and rolling a 6 on the dice.
Tree diagrams help to visualise the question and allow you to calculate a number of probabilities for a variety of scenarios very quickly.
Tree Diagrams - Introduction
Suppose that you have to pick two of the shapes from those below but you have to replace the first shape before you pick the second shape.
Well you could draw a table showing all the different possible combinations that you could possibly have. The problem with this is that it could take some time to do.
An alternative to this would be to draw a tree diagram.
Constructing a tree diagram
When you pick a first shape the options are that of a square, triangle and a circle.
The probabilities of these as single events would be:
P(\text { square })=\frac{3}{6} ; P(\text { triangle })=\frac{2}{6} \text { and } P(\text { circle })=\frac{1}{6}This can be shown on a tree diagram as follows:
The above shows the probabilities for the first picking. Because the first shape is replaced, when it comes to picking the second shape you still have the option of picking a square, triangle or a circle.
This can be shown by adding more branches to the tree diagram as shown below. (When drawing tree diagrams make sure that you give yourself plenty of space in which to work and also draw your diagrams clearly and big)
Tree Diagrams - Larger Ones
Calculating the Probabilities
Once you have drawn the tree diagram it is then possible to calculate the probabilities and this is done simply by following the branches of the tree and then multiplying the probabilities together.
For instance to find the probability of picking two squares on the both attempts (using the above tree diagram) we have:
P(\text { two squares })=\frac{3}{6} \quad \times \frac{3}{6}=\frac{9}{36}And if you wanted to find the probability of picking a circle followed by a triangle just follow the branches again and multiply the appropriate probabilities as follows:
P(\text { circle then triangle })=\frac{1}{6} \times \frac{2}{6}=\frac{2}{36}
This is a very brief introduction to the use of tree diagrams as the questions can be more complicated such as when something is not replaced, the probabilities change, and if doing higher GCSE Maths, then there are some questions that do involve the use of algebra.
When you are revising tree diagrams as a part of your probability revision for GCSE maths, it is important that you are drawing your tree diagram clearly as well as writing all probabilities clearly as well.
There will be instances where you will be given a tree diagram and are asked to complete it by including the correct fractional values, but if one is not provided, then you should be prepared to draw one.
If you are doing the GCSE higher paper then you can encounter probability questions that require the use of algebra. We will be creating an article based on this topic so be sure to read to it.