Tree Diagram

A tree diagram is used to summarize the probabilities associated with a sequence of random events. The set of branches emanating from any given start point represent all the possible events that could follow. Each branch is labeled according to the probability of the event occurring, given the events that have previously happened. Hence, the sum of probabilities from each set of branches must equal 1.

Each path from the start of the tree to the end defines an outcome in the sample space. The outcomes defined by the paths are mutually exclusive. All outcomes in the sample space are represented by a path. The probability of each outcome is obtained by multiplying the conditional probabilities along the path, which is often called the “Multiplication Rule.” Therefore, these probabilities sum to 1.

An interesting application of tree diagrams is provided by the following example, toward anonymizing a survey question. Consider the following:

• Toss a fair coin.
• If you get a “head,” answer the question:
  • Have you ever cheated in an exam? Yes or No.
• If you get a “tail,” answer the question:
  • Flip coin a second time. Did you get a head on the second flip? Yes or No.

It is not possible to tell whether a particular individual answered “Yes” because they cheated in an exam or got a head on the second flip. However, if we have a large sample of responses, we can estimate the proportion of people who have cheated in an exam.

Figure 1 Tree Diagram for Anonymized Exam Cheating Question

Consider the tree diagram in Figure 1, created in PowerPoint.

The probability of answering “Yes” is given by

This calculation assumes a large sample with fair coins to minimize bias in the coin flip probabilities. If 35% of sample answered “Yes,” then

which can be rearranged to give

The probability that someone cheated in an exam is independent of the first head, so P(Cheat|Head) = P(Cheat). Hence, from the survey, we estimate that around 20% of students have cheated in an exam.