Short on time? This page is the recap. For explanations, a Venn diagram and worked examples, use the full study guide.
Recap
- The intersection A ∩ B is the set of outcomes in both A and B. Read it “A and B”.
- The joint probability P(A ∩ B) is the probability that A and B both happen on the same trial.
- Two events are mutually exclusive (or disjoint) if they cannot happen at the same time.
- So mutually exclusive events have P(A ∩ B) = 0. Any joint probability above 0, however small, means the events are not mutually exclusive.
Key relationships
| Idea | How to find or use it |
|---|---|
| Joint probability, equally likely outcomes | P(A ∩ B) = (outcomes in both A and B) ÷ (total outcomes) |
| Joint probability from a two-way table | P(row ∩ column) = cell count ÷ grand total |
| Mutually exclusive | P(A ∩ B) = 0; on a Venn diagram, no overlap with outcomes in it |
| Not mutually exclusive | P(A ∩ B) > 0; at least one outcome is in both events |
| Event and its complement | always mutually exclusive |
| Two rows (or two columns) of one two-way table | always mutually exclusive: each individual is in one row and one column |
Assumptions and conventions
- Both events refer to the same trial (the same roll, the same randomly chosen person or ticket).
- A two-way table is used for one individual chosen at random from the table.
- A zero cell shows P(A ∩ B) = 0 for an individual chosen from that data set. It does not, on its own, prove the combination is impossible in a wider population.
Mistakes to avoid
- Calling small joint probabilities “mutually exclusive”. Only 0 counts.
- Thinking mutually exclusive means complement. Disjoint events can leave outcomes in neither event.
- Dividing a cell by a row or column total when you want a joint probability. Use the grand total.
- Mixing up “cannot happen together” with “does not affect each other”. The second idea is independence (Topic 2.7).
- No justification. Always show P(A ∩ B), compare it with 0 and conclude in context.
Quick self-check
- A card is drawn at random from cards numbered 1 to 20. Are “multiple of 9” and “even and greater than 15” mutually exclusive? (No: 18 is in both, so P(both) = 1/20 = 0.05 > 0.)
- For two events, P(A ∩ B) = 0. What can you conclude? (A and B are mutually exclusive: they cannot happen on the same trial.)
- In a survey of 80 shoppers, none both paid by cash and used self-checkout. For one shopper chosen at random from these 80, what is P(cash ∩ self-checkout)? Does this prove no shopper ever does both? (0/80 = 0, so the events are mutually exclusive for these 80 shoppers. It does not prove it for all shoppers: the combination may just not have appeared in this sample.)
Next: practice questions.