Concept

Tree Diagram

A tree diagram illustrates the set of all possible outcomes of an experiment involving several stages. It is formed by three principle parts.

Nodes: Each node represents a certain event.
Branches: A branch connects two nodes. Several branches can extend from each node.
Probabilities: The probability of each outcome is written on its corresponding branch.

Tree diagrams help visualize the probability of events. They can also be used for finding all possible arrangements of a set of elements. Because this type of diagram is commonly used in probability, they are also called probability trees or probability tree diagrams.

Example: Making a Tree Diagram

Consider the experiment of flipping a fair coin and then rolling a fair die.

To make the tree diagram, first, the experiment's stages need to be identified. In doing so, include the possible outcomes and probabilities.

Stage $1 .$ Flipping the coin.
- The possible outcomes are heads and tails.
- Each outcome has a probability of $\frac{1}{2} .$
Stage $2 .$ Rolling the die.
- The possible outcomes are $1,$ $2,$ $3,$ $4,$ $5,$ and $6 .$
- Each outcome has a probability of $\frac{1}{6} .$

Beginning at a root node, two branches extend connecting to the nodes that represent the possible outcomes of the first stage — heads and tails. Then, six branches are extended from each of these nodes to connect with the possible outcomes of the second stage —

1,

2,

3,

4,

5,

and

6 .

Each branch must be labeled with its probability.

Tree diagram of the flipping of a fair coin and the rolling of a fair die

Using the diagram, the probability of any event can be calculated by multiplying the probabilities of the connected branches. For example, the probability of the coin landing on heads and then rolling a

3

is given by the product of

\frac{1}{2}

and

\frac{1}{6} .

P (head and 3) = \frac{1}{2} \cdot \frac{1}{6} ⇕ P (head and 3) = \frac{1}{1 2}

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Example: Making a Tree Diagram