Tree Diagrams

Understanding Tree Diagrams

• A Tree Diagram is a useful graphical representation to visualize the different possible outcomes of a series of events.
• Each branch of the tree represents an outcome, while each node or split represents an event.
• From a starting point, lines branch out depicting possible outcomes, hence the name ‘Tree Diagram’.
• A tree diagram specifically shows all outcomes and stages of a process in a clear, easy to follow layout.

Using Tree Diagrams for Probability

• Tree diagrams are particularly useful in probability because they record all possible outcomes in a clear and systematic way.
• Each branching section or node of the tree diagram represents an independent event.
• It helps in the calculation of the probability of outcomes for compound events, particularly if events are dependent.

Constructing a Tree Diagram

• When constructing a Tree Diagram, start by identifying the first event and list all the possible outcomes. Draw a line stemming from a point for each of these outcomes.
• For each line, consider the subsequent event and its possible outcomes. Draw a line for each of these, branching from the endpoint of the first line.
• This process is repeated until all outcomes of all events have been explored.
• The probabilities of the outcomes are usually written on the branches, while the outcomes themselves are written at the end of the branches.

Interpreting a Tree Diagram

• Once a tree diagram is complete, it can be used to calculate the probability of certain outcomes.
• To calculate the probability of a series of events, multiply the probabilities along the branches of the tree that represent these events.
• Tree diagrams also clearly illustrate the concept of independent events, as the branches at each node are not influenced by the outcome of previous branches.
• In contrast, dependent events can be illustrated by probabilities on branches changing depending on the outcomes of previous events.

Example of a Tree Diagram

• For example, consider a bag containing three red balls and two blue balls. A ball is drawn from the bag, its colour noted, and then it is replaced before a second ball is drawn. A tree diagram can be used to represent the possible outcomes and their probabilities.
• The first node has two branches - one for drawing a red ball first (with a probability of 3/5) and one for drawing a blue ball first (with a probability of 2/5).
• From each of these branches extend two more - one for drawing a red ball second and one for drawing a blue ball second.
• The four final branches of the tree represent the four possible outcomes: red then red, red then blue, blue then red, blue then blue.
• The probability of each of these outcomes can be calculated by multiplying along the branches. For example, the probability of drawing a red ball then a blue ball is (3/5) * (2/5) = 6/25.