Maths › Statistics › Probability and Venn diagrams
Probability and Venn diagrams
Probability is careful counting, written formally. Venn diagrams keep the counting accurate, the addition rule handles events that overlap, and two special cases, mutually exclusive and independent, account for most of what the exam asks.
Builds on Sampling and the large data set.
IN THIS TOPIC
- Represent events on a Venn diagram, including three sets, 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.
COMMON MISCONCEPTION
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 a rectangle holding everything that can happen. Every region carries a probability and the regions total 1. Reading the diagram is then arithmetic. A ∩ B, “and”, is the overlap. A ∪ B, “or”, is everything inside either circle. The complement A′, “not A”, is everything outside A.
Fill diagrams from the inside out. Overlap first, then the “only” regions by subtraction, then the outside so the total reaches 1. Almost every Venn error is the same one: writing P(A) = 0.5 into the A-only region and forgetting that P(A) already includes the overlap.
Three-circle diagrams follow the same discipline and punish impatience harder. Start with the central region where all three meet, then the three pairwise slivers, then the three singles, then the outside. Work outwards and every unknown you meet has only one equation attached to it.
The addition rule
Adding P(A) to P(B) counts the overlap twice, so it comes off once. The rule rearranges freely and the exam's favourite direction is backwards, hunting 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.
Show the comparison and then write the verdict as a sentence. Bare arithmetic with no conclusion is the standard way to lose the last mark here.
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, and the test is P(A ∩ B) = P(A) × P(B). The two ideas are different and very nearly opposite. Two mutually exclusive events with non-zero probabilities are strongly dependent, because seeing one happen tells you at once that the other did not. To show independence in an exam, compute both sides of the multiplication test and compare them out loud.
ASSESSMENT FOCUS
- Fill Venn diagrams from the overlap outwards and make the regions total 1.
- The addition rule is the one you will use most. Expect to rearrange it for the overlap far more often than you use it forwards.
- Independence is a calculation. Show P(A ∩ B) and P(A) × P(B), then say whether they match.
- Never write “independent” when you mean “mutually exclusive”. Examiners read both words literally and mark them 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 total 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.
Or read them with their worked answers on the probability and venn diagrams questions page.
CHECK YOUR PROGRESS
Rate how confident you feel with each objective for this lesson. Ratings are saved in this browser, on this device, unless you sign in.
- Represent events on a Venn diagram, including three sets, 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.
Open the full revision checklist to see every objective in the course in one place.