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Tree Diagram

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Tree Diagram

A tree diagram illustrates the set of all possible outcomes of an experiment involving several stages. A tree diagram is formed by three main parts.
  • Nodes. Each node represents a certain event.
  • Branches. A branch connects two nodes. Each node can have several branches.
  • Probabilities. The probability of each outcome is written on its corresponding branch.

Tree diagrams can 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, tree diagrams are also known as a probability trees or probability tree diagrams.


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

A fair coin and a fair die

To make the tree diagram, the stages need to be identified first.

  • Stage Flipping the coin.
    • The possible outcomes are heads and tails.
    • Each outcome has a probability of
  • Stage Rolling the die.
    • The possible outcomes are and
    • Each outcome has a probability of
From a root node two branches extend to connect with the nodes that represent the possible outcomes of the first stage, which are heads and tails. Then, six branches are extended from each of these nodes to connect with the possible outcomes of the second stage, which are and While doing this process, 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 heads and then being rolled is given by the product of and