For A or B, add the probabilities and then subtract any overlap, because outcomes in both events have been counted twice. If the events cannot happen together, the overlap is zero and you only add.
This lesson finishes the combined-event probability section. It builds on using tree diagrams with and without replacement.
How do I tell whether events are mutually exclusive?
Ask whether any single outcome can belong to both events. If one can, the events overlap and are not mutually exclusive.
List the outcomes of each event. A shared outcome is an overlap, and it must be counted only once.
Worked example: one sample space, two pairs
A card is chosen at random from 20 cards numbered 1 to 20.
Pair 1: mutually exclusive. Event C: a number less than 4, which gives {1, 2, 3}, so 3 outcomes. Event D: a number greater than 17, which gives {18, 19, 20}, so 3 outcomes.
No number is in both lists, so P(C or D) = 3/20 + 3/20 = 6/20 = 3/10.
Pair 2: not mutually exclusive. Event A: a multiple of 3, which gives {3, 6, 9, 12, 15, 18}, so 6 outcomes. Event B: an even number, which gives 10 outcomes: {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}.
The numbers 6, 12 and 18 are in both lists, so there are 3 overlapping outcomes.
P(A or B) = 6/20 + 10/20 − 3/20 = 13/20.
Check by listing A or B directly: {2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20}. That is 13 outcomes, so P = 13/20.
The mistake that costs marks
The slip is to add 6/20 and 10/20 and stop, giving 16/20. The working looks tidy, but it counts 6, 12 and 18 twice.
The check that catches it is the list. Count the outcomes in your list: there are only 13 distinct numbers, so 16/20 cannot be right. Whenever you add, ask whether the events overlap.
| Step | Wrong | Right |
|---|---|---|
| P(A) + P(B) | 6/20 + 10/20 = 16/20 | 6/20 + 10/20 = 16/20 |
| Overlap | Ignored | Subtract 3/20 |
| Answer | 16/20 | 13/20 |
Check yourself
In a class of 30 students, 18 play football, 12 play badminton and 5 play both. A student is chosen at random. Find P(football or badminton) and P(neither).
Answer
The events overlap, because 5 students play both. P(football or badminton) = 18/30 + 12/30 − 5/30 = 25/30 = 5/6.
Neither means not in either group. The students who play at least one sport are 25, so 30 − 25 = 5 play neither, and P(neither) = 5/30 = 1/6.
Check: 5/6 + 1/6 = 1, as expected for an event and its complement.
What to study next
Put all four skills together on the combined-event probability practice set. If the tree diagram step still slows you down, return to using tree diagrams with and without replacement.
For a teacher to go through your working, see online one-to-one Mathematics tuition.