Probabilities of Combined Events using Tree Diagrams (College Board AP® Statistics)
Study Guide
Written by: Dan Finlay
Reviewed by: Lucy Kirkham
Tree diagrams
What is a tree diagram for probability?
A tree diagram is a way to visualize probabilities and conditional probabilities when dealing with combined events
Tree diagrams show how the occurrence of an event affects the probability of another event occurring
The first set of branches is labeled with the probabilities of mutually exclusive events
This is typically just a given event and its complement
e.g. rolling a 6 and not rolling a 6
However, there can be more than two events
e.g. rolling a 1, rolling a 2, rolling a 3, etc.
The probabilities on these branches add up to 1
The second sets of branches are labeled with the conditional probabilities given that one of the previous events has occurred
The conditional probabilities on the branches for each set of events add up to 1
Tree diagrams can have more than two sets of events
However, in these cases, there are more efficient ways to find the probabilities
See the student guide on finding probabilities of combined events using the rules
Examiner Tips and Tricks
Be careful with the probabilities on the second set of branches, these are conditional probabilities. In the above example, is the probability that a person wins a game given that they rolled a six. It is not the probability that the person wins the game.
How can I find probabilities from a tree diagram?
You can find the probability of the intersection of two events by multiplying the probabilities on their branches
This is just the multiplication rule
To find the probability of an event in the second set of branches:
Identify any intersections which involve that event
Add together the probabilities of those intersections
e.g.
It can be quicker to find the probability of the complement of an event
If you are asked to find the probability that or occurs
find the probability that neither nor occurs
then subtract this from 1
How can I find conditional probabilities from a tree diagram?
Suppose the tree diagram has the events and on its first set of branches and the events and on its second set
e.g. let be the event that a 6 is rolled and be the event that the game is won
The conditional probabilities , , and are labeled on the second set of branches
e.g. the probability that the game is won given that a six is rolled is labeled on the relevant branch
The conditional probabilities , , and can be found using the conditional probability formula
e.g. you have to use this formula to find the probability that a six is rolled given that the game is won
Remember you can partition the event using and
Worked Example
To get to work, Jen either rides the subway or takes a cab. The probability that Jen is late for work depends on how she travels there. An incomplete tree diagram of the possible outcomes for Jen being late to work or not is shown below.
(a) Complete the tree diagram by writing the correct probability in each of the six empty boxes.
Answer:
The probabilities on each pair of branches add up to 1
Calculate the three missing probabilities on the branches
The probability of each outcome is found by multiplying the probabilities on the branches that lead to that outcome
Calculate the three missing probabilities for the outcomes
(b) Calculate the probability that Jen is late for work.
Answer:
Add together the probabilities of the outcomes that involve Jen being late for work
(c) Given that Jen is late for work, find the probability that she rode the subway.
Answer:
The conditional probability required is
Use the conditional probability formula and the answer from the previous part
Given that Jen is late for work, the probability that she rode the subway is 0.223
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