Probability and statistics · GCSE Maths
Venn diagrams and probability
GCSE Maths Venn diagrams: use P(A ∪ B) = P(A) + P(B) − P(A ∩ B), fill the overlap first, and read “and”, “or” and “given that” from the regions.
Fill the intersection first. AND is the overlap. OR is everything in either circle, which is P(A) + P(B) minus the overlap so it is not counted twice.
The important bits
What you need to know
- 1
P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Union is OR; intersection is AND. Subtract the overlap because it was in both P(A) and P(B).
- 2
If A and B are mutually exclusive they cannot happen together, so P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B). Disjoint circles, no overlap.
- 3
If A and B are independent, P(A ∩ B) = P(A) × P(B). Independence is a multiplication rule, not a picture of separate circles.
- 4
On a Venn diagram of frequencies, put the AND number in the overlap first, then the rest of A, then the rest of B, then outside both. The four regions must sum to the total.
- 5
P(A | B) means given that B has happened: restrict to the B circle, then P(A | B) = P(A ∩ B) / P(B). That is the overlap divided by the whole of B.
- 6
P(not A) is the complement, 1 − P(A), the region outside A including outside both circles if you are in the universal set.
- 7
Two-way tables and Venn diagrams hold the same four numbers: A and B, A not B, B not A, neither. Convert if the question is easier as a table.
- 8
Probabilities from a Venn of frequencies: divide each region by the total. Do not divide by the size of one circle unless the question is conditional.
Quotations worth analysing
Short evidence. Real method.
“P(A ∪ B) = P(A) + P(B) − P(A ∩ B)”
The minus is the method. Adding P(A) and P(B) without it double-counts the overlap and can produce a probability greater than 1.
“Fill in the intersection first”
If 8 students do both, 8 goes in the overlap. Then 20 who do French become 12 in French only. Starting with 20 in the whole French circle on top of the 8 double-counts.
“P(A given B) = P(A ∩ B) / P(B)”
The sample space shrinks to B. Use the overlap over the whole B region, not over the universal total, unless B is the whole set.
Go deeper
Overlap first, then the rest of each circle
In a class of 30, 18 do French, 12 do Spanish, 7 do both. The 7 is the intersection: write 7 in the overlap. French only is 18 − 7 = 11. Spanish only is 12 − 7 = 5. Neither is 30 − 11 − 7 − 5 = 7. Those four numbers 11, 7, 5, 7 add to 30. P(French and Spanish) = 7/30. P(French or Spanish) = (11 + 7 + 5)/30 = 23/30, which matches 18/30 + 12/30 − 7/30. P(only French) is 11/30, not 18/30. “Only” is a word that means the crescent, not the whole circle. If a student is chosen at random, P(Spanish | French) = 7/18, because you are already inside the French circle of 18, and 7 of those also do Spanish. Using 7/30 here ignores the “given French”.
Go deeper
Union, independence and exclusive are three different ideas
Mutually exclusive: the overlap is empty. A card cannot be a heart and a spade. Then OR is just add. Independent: knowing A happened does not change P(B), so P(A ∩ B) = P(A)P(B). Two coins are independent and not exclusive: they can both be heads. Exclusive events with non-zero probability cannot be independent, because if A has happened, B cannot, so P(B | A) = 0 ≠ P(B). Students mash the words: they multiply because they see two circles, or they add because they see the word “and”. AND is overlap (multiply only if independent, or read the overlap from the diagram). OR is union (add and subtract the overlap unless exclusive). The Venn does not assume independence; it just shows the four regions. Use the numbers in the regions rather than a slogan.
Go deeper
Three-set Venns and “given that”
Three circles have a central triple overlap. Fill that first, then each pairwise overlap (minus the triple), then the only-one crescents, then outside. The same “subtract what you have already placed” discipline applies. Conditional probability is a restricted total: given that the student does music, only the music circle is the new whole, including all overlaps that sit inside music. If 40 students, 10 in the music circle, 4 of those also do drama, P(drama | music) = 4/10. A question that gives P(A), P(B) and P(A ∪ B) and asks for P(A ∩ B) is the addition formula rearranged: P(A ∩ B) = P(A) + P(B) − P(A ∪ B). That algebra is the Venn with the labels missing. Check the result is between 0 and the smaller of P(A) and P(B); an intersection cannot be larger than either set.
See the idea in action
P(A) = 0.6, P(B) = 0.5, P(A ∩ B) = 0.3. Find P(A ∪ B) and P(A | B). Step 1: P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 0.6 + 0.5 − 0.3 = 0.8. Step 2: Regions: only A is 0.6 − 0.3 = 0.3; overlap 0.3; only B is 0.5 − 0.3 = 0.2; neither is 1 − 0.8 = 0.2. Step 3: Check: 0.3 + 0.3 + 0.2 + 0.2 = 1. Step 4: P(A | B) = P(A ∩ B) / P(B) = 0.3 / 0.5 = 0.6. If they had been independent, P(A ∩ B) would have been 0.6 × 0.5 = 0.3, which matches this particular example, so these events happen to be independent as well. That is a check, not a rule you assume at the start.
Exam technique
Turn knowledge into marks
Write the four region values (only A, both, only B, neither) before you answer. Conditional questions divide by the new total, which is one circle, not the whole rectangle.
Common mistakes
Do not give these marks away
- 01
Using P(A or B) = P(A) + P(B) when A and B can overlap, so the intersection is counted twice.
- 02
Putting P(A) in the whole of circle A without subtracting the overlap that has already been placed.
- 03
Calculating P(A | B) as P(A ∩ B) divided by the universal total instead of by P(B).
P(A) = 0.6, P(B) = 0.5, P(A ∩ B) = 0.3. P(A ∪ B) is
A0.8
B1.1
C0.3
D0.15
Show the answer
0.8. 0.6 + 0.5 − 0.3 = 0.8. 1.1 is the double-count without subtracting the overlap. 0.3 is the intersection. 0.15 would be a mistaken extra product.
Quick questions
If this is the bit you searched
What is the formula for P(A or B)?
P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B cannot happen together, the intersection is 0 and you just add.
How do you fill in a Venn diagram?
Put the AND frequency in the overlap first, subtract it from each total to get the “only” regions, then subtract all three inner regions from the universal total to get neither.
How do you find a conditional probability from a Venn?
P(A | B) = (overlap) / (whole of B). Restrict the sample space to B, then see what fraction of that also sits in A.
Are independent events the ones with no overlap?
No. No overlap means mutually exclusive. Independent events can overlap; their intersection probability is the product P(A)P(B).