Tree Diagrams (DP IB Applications & Interpretation (AI)): Revision Note

Tree Diagrams

What is a tree diagram?

• A tree diagram is another way to show the outcomes of combined events

  • They are very useful for intersections of events

• The events on the branches must be mutually exclusive

  • Usually they are an event and its complement

• The probabilities on the second sets of branches can depend on the outcome of the first event

  • These are conditional probabilities

• When selecting the items from a bag:

  • The second set of branches will be the same as the first if the items are replaced

  • The second set of branches will be the different to the first if the items are not replaced

How are probabilities calculated using a tree diagram?

• To find the probability that two events happen together you multiply the corresponding probabilities on their branches

  • It is helpful to find the probability of all combined outcomes once you have drawn the tree

• To find the probability of an event you can:

  • add together the probabilities of the combined outcomes that are part of that event

    • For example: 

  • subtract the probabilities of the combined outcomes that are not part of that event from 1

    • For example: 

Do I have to use a tree diagram?

• If there are multiple events or trials then a tree diagram can get big

• You can break down the problem by using the words AND/OR/NOT to help you find probabilities without a tree

• You can speed up the process by only drawing parts of the tree that you are interested in

Which events do I put on the first branch?

• If the events A and B are independent then the order does not matter

• If the events A and B are not independent then the order does matter

  • If you have the probability of A given B then put B on the first set of branches

  • If you have the probability of B given A then put A on the first set of branches