Tree Diagrams

Understanding Tree Diagrams

  • Tree Diagrams are graphical representations used to map out outcomes and their probabilities for sequential events.
  • They are called “tree diagrams” due to their branching structure that visualises everything from roots to leaves, resembling a tree.
  • When working with probability, tree diagrams help visualise all possible outcomes clearly for complex statistical problems.

Constructing Tree Diagrams

  • Begin by drawing a point or a node that represents the start of your event.
  • Branch out from the starting point, each branch representing a possible outcome. Remember that the sum of probabilities along each pathway should add up to 1.
  • For sequential events, a second set of branches represents the outcomes of the second event. These branches should extend from the ends of the first set of branches.
  • Continue this process for as many events as necessary, making sure each new set of branches extends from the endpoint of the previous set.
  • Each complete pathway from the start to finish represents one possible series of outcomes for the events.

Interpreting Tree Diagrams

  • Each branch represents a different possible outcome. The length or size of the branch does not affect probability.
  • The probability of each outcome is typically written on the respective branch. The combined probability of a sequence of outcomes is the product of their individual probabilities.
  • If two events are independent, the probability of the second event does not change with different outcomes of the first event. This is reflected on a tree diagram by having the same set of branches extend from each end of the first set of branches.
  • If two events are dependent, the probabilities on the second set of branches will change depending on the outcome of the first event.

Using Tree Diagrams for Problem Solving

  • Tree diagrams become increasingly useful for complex problems with multiple stages or dependent events.
  • To find the probability of a sequence of events, find the specific path in your tree diagram that represents the outcomes in that sequence. Then, multiply the probabilities along that path.
  • Adding probabilities is applicable when finding the aggregate probability of multiple unrelated outcomes.
  • The sum of all probabilities in a tree diagram should equal to 1, if they do not, an outcome may have been overlooked.