Probability of Combined Events
How to Use a tree diagram for combined events
Use a tree diagram to find the probability of two or more events happening together.
Before you start
- Writing a probability as a fraction
- Multiplying two fractions
- Adding fractions with the same denominator
- Knowing the probabilities of one event's outcomes add up to 1
When to use it
Use a tree diagram to find the probability of two or more events happening together.
The steps
- Draw a branch for each outcome of the first event and label each with its probability.
- From each branch, draw the outcomes of the second event.
- For dependent events, reduce the probabilities on the second set of branches.
- Multiply along a path to find the probability of that combination.
- Add the probabilities of all paths that satisfy the question.
Worked example
A bag holds 3 red and 2 green marbles. A marble is drawn, its colour noted, then put back; a second marble is then drawn.
Find the probability that the two marbles are the same colour.
- Draw a branch for each outcome of the first draw: Red with probability 3/5 and Green with probability 2/5.
- From each first branch, draw the second-draw outcomes: Red and Green again.
- The marble is replaced, so the bag is unchanged and the events are independent, the second probabilities stay 3/5 and 2/5 (no reducing).
- Multiply along each 'same colour' path: Red-Red = 3/5 × 3/5 = 9/25; Green-Green = 2/5 × 2/5 = 4/25.
- Add the paths that give the same colour: 9/25 + 4/25 = 13/25.
A second example, with a twist
The marble is not replaced, so the events are dependent and the second-draw probabilities must be reduced. A bag holds 4 red and 3 yellow marbles.
Two marbles are drawn one after the other without replacement. Find the probability that both are the same colour.
- Draw branches for the first draw: Red 4/7 and Yellow 3/7 (total 7 marbles).
- From each, draw the second-draw branches for Red and Yellow.
- There is no replacement, so reduce the totals: after a red, 3 red remain out of 6, so Red-Red uses 3/6; after a yellow, 2 yellow remain out of 6, so Yellow-Yellow uses 2/6.
- Multiply along each 'same colour' path: Red-Red = 4/7 × 3/6 = 12/42; Yellow-Yellow = 3/7 × 2/6 = 6/42.
- Add the same-colour paths: 12/42 + 6/42 = 18/42 = 3/7.
Formulae you may need
Formula pages
Practise this in a KBAT problem
Frequently asked questions
When do I multiply and when do I add on a tree diagram?
Multiply along a single path, because that is one event AND the next both happening. Add across different paths, because that is one path OR another satisfying the question.
So 'and' means multiply, 'or' means add.
How do I know if the events are dependent?
Ask whether the first outcome changes what is left. If there is no replacement, fewer items remain, so the second probabilities change, the events are dependent and you must reduce the totals.
With replacement, nothing changes and the events are independent.
Do the branches from one point have to add up to 1?
Yes. From any single point on the tree, the probabilities on the branches leaving it should total 1, because they cover every possible outcome of that event.
It is a quick way to check you have not made a slip before multiplying.