Maths › Statistics › Probability and Venn diagrams
Probability and Venn diagrams
Probability is careful counting dressed up. Venn diagrams keep the counting honest, one addition rule covers every overlap, and two special cases, mutually exclusive and independent, do most of the exam's work.
Builds on Sampling and the large data set.
IN THIS TOPIC
- Represent events on a Venn diagram and read probabilities from its regions.
- Use P(A ∪ B) = P(A) + P(B) − P(A ∩ B) in both directions.
- Test events for mutual exclusivity and for independence, and keep the two ideas apart.
WHAT YOU PROBABLY THINK
Mutually exclusive events are independent, since they have nothing to do with each other.
Events as regions
An event is a set of outcomes, and a Venn diagram draws events as circles inside the rectangle of everything that can happen. Every region gets a probability and the regions sum to 1. Reading the diagram is then arithmetic: A ∩ B (“and”) is the overlap, A ∪ B (“or”) is everything inside either circle, and the complement of A (“not A”) is everything outside A.
Fill diagrams from the inside out: the overlap first, then the “only” regions by subtraction, then the outside so the total reaches 1. Most Venn errors are filling P(A) = 0.5 into the A-only region and forgetting it includes the overlap.
The addition rule
Adding P(A) and P(B) counts the overlap twice, so subtract it once. The rule rearranges freely, and the exam's favourite direction is finding the overlap:
WORKED EXAMPLE
Finding the overlap, then testing independence
P(A) = 0.5, P(B) = 0.4 and P(A ∪ B) = 0.7. Find P(A ∩ B) and determine whether A and B are independent.
Rearranging: P(A ∩ B) = 0.5 + 0.4 − 0.7 = 0.2.
Independence test: P(A) × P(B) = 0.5 × 0.4 = 0.2, which equals P(A ∩ B), so A and B are independent.
The conclusion sentence matters: state the comparison and the verdict, never the arithmetic.
Two special cases
Mutually exclusive events cannot both happen: P(A ∩ B) = 0, the circles are drawn apart, and the addition rule collapses to P(A ∪ B) = P(A) + P(B).
Independent events carry no news about each other: P(A ∩ B) = P(A) × P(B). The two ideas are different and nearly opposite: mutually exclusive events are strongly dependent, because seeing one happen tells you the other did not. To show independence in an exam, compute both sides of the multiplication test and compare them explicitly.
THE EXAM BIT
- Fill Venn diagrams from the overlap outwards, and make the regions sum to 1.
- The addition rule is the workhorse; expect to rearrange it for the overlap.
- Independence is a calculation, not a feeling: show P(A ∩ B) and P(A) × P(B) and say whether they match.
- Never call events “independent” when you mean “mutually exclusive”; the examiner reads both words literally.
CHECK YOURSELF
P(A) = 0.6, P(B) = 0.3, and A and B are mutually exclusive. Write down P(A ∩ B) and find P(A ∪ B).
Show a hint
Mutually exclusive fixes the overlap immediately.
Show the answer
P(A ∩ B) = 0 by definition of mutually exclusive.
P(A ∪ B) = 0.6 + 0.3 − 0 = 0.9.
Draw the diagram, fill the overlap first, and make everything sum to one.
Exclusive means the overlap is zero; independent means the overlap equals the product. The two are almost never both true.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
Rate how confident you feel with each objective for this lesson. Ratings are saved in this browser, on this device only.
- Represent events on a Venn diagram and read probabilities from its regions.
- Use P(A ∪ B) = P(A) + P(B) − P(A ∩ B) in both directions.
- Test events for mutual exclusivity and for independence, and keep the two ideas apart.
No animated video for this topic yet; these notes stand alone.