Tree Diagrams

Purpose of Tree Diagrams

Tree diagrams are a visual tool used to represent sequential events in probability. They help to:

• Show all possible outcomes of a sequence of events clearly
• Visualise the structure of combined events step-by-step
• Calculate the probability of combined events by following branches

Each branch in the tree represents an event or outcome, and by tracing along the branches, you can find all possible sequences of outcomes.

Constructing Tree Diagrams

To construct a tree diagram:

• Start with a single point (root) representing the start before any events
• Draw branches from this point for each possible outcome of the first event
• From the end of each branch, draw further branches for the outcomes of the next event
• Label each branch with the probability of that outcome occurring

The probabilities on branches must be between 0 and 1, and for each set of branches from a single point, the probabilities should add up to 1.

To find the probability of a combined sequence of events, multiply the probabilities along the branches of that path.

For instance, if the first event has two outcomes with probabilities 0.6 and 0.4, and the second event following the first outcome has probabilities 0.5 and 0.5, then the combined probability of the first outcome followed by the first outcome of the second event is:

$$0.6 \times 0.5 = 0.3$$

Using Tree Diagrams for Multiple Events

Tree diagrams are especially useful when dealing with multiple events that happen one after another. These events can be:

• Independent events: The outcome of one event does not affect the probabilities of the next event.
• Dependent events: The outcome of one event affects the probabilities of the next event.

When events are independent, the probabilities on branches remain the same regardless of previous outcomes. When dependent, probabilities may change based on earlier results and must be adjusted accordingly.

To find the total probability of an event that can happen in different ways, sum the probabilities of all branches that lead to that event.

For example, if you want the probability of getting at least one head when tossing a coin twice, you add the probabilities of all branches where there is at least one head.

This method ensures you consider every possible outcome and calculate probabilities accurately.

Example: A bag contains red and blue balls. You pick one ball, note its colour, then pick a second ball without replacing the first. The probability of picking a red ball first is $\frac{3}{5}$, and blue is $\frac{2}{5}$. If the first ball is red, the probability of picking a red ball second is $\frac{2}{4}$ (since one red ball is removed). If the first ball is blue, the probability of picking a red ball second is $\frac{3}{4}$. The tree diagram shows these branches with probabilities labelled. To find the probability of picking two red balls, multiply along the branches: $\frac{3}{5} \times \frac{2}{4} = \frac{6}{20} = \frac{3}{10}$.