Understanding Tree Diagrams

Tree diagrams are graphical representations of a series of probabilistic experiments.
A tree diagram starts with a single point, branches out to show all the possible outcomes of an event, and further branches out to show the possible outcomes of a second event, and so on.
Each point that causes a split into different branches is referred to as a node.
Each branch represents a possible outcome.

Drawing Tree Diagrams

Start with a point on the left and branch out to the right for all the possibilities of the first event. Label each branch with the probability of that outcome.
From each outcome of the first event, draw branches to represent possible outcomes of the second event. Again, label each branch with the probability of that outcome.
Continue in this way for each successive event. Put all the final outcomes of the experiment at the end of the branches.

To find the overall probability of an outcome, multiply the probabilities along the branches of the tree that lead to that outcome.
The sum of the probabilities from a node must equal one. This often helps when working out missing probabilities.

These diagrams can be used to illustrate independent events, where the outcome of the first event does not affect the outcome of the second event. In this case, the tree branches look symmetrical.
The tree diagrams will also represent dependent events, where the outcome of the first event affects the outcome of the second event. In this case, the probabilities on each branch could be different.

They provide a visual representation of all possible outcomes and associated probabilities, making it easier to understand complex probabilistic situations.
By following paths through the diagram, you can work out the probabilities of particular combinations of outcomes.
Predicting probabilities is easier when you can see all the outcomes and their chances.