01 · Check totals
Recover the art and music totals from the four regions.
Hint
Add each only region to the overlap.
Worked solution
Art: 18+12 = 30. Music: 22+12 = 34. Whole sample: 18+12+22+28 = 80.
Understand · explore · practise
Two-event Venn diagrams
Build a two-event Venn diagram, interpret intersections and inclusive unions, and find only, neither and exactly-one probabilities without double counting.
Before you startEvents, complements and fractions.
01 / Separate four regions
Every pupil belongs to exactly one region.
Intersection A ∩ M means both. Union A ∪ M means A or M or both.
The rectangle represents the whole stated sample space.
The two circles overlap because a pupil may study both subjects. The circles are not drawn with areas proportional to their probabilities; use the labelled counts.
Select an eventExplore
Uniformly choose one of 80 pupils. Let A mean studying art and M mean studying music. Region counts: art only 18; both 12; music only 22; neither 28.
Art includes art only and both: (18+12)/80 = 3/8.
A tick marks each included count. A prime (′) means complement relative to these 80 pupils.
02 / Put the overlap in first
A set total includes the intersection.
Watch: subtract overlap before filling the outside
Pause, replay or seek freely. The notes explain the same idea and stay in view.
Of 80 pupils, 30 study art, 34 study music and 12 study both.Worked example
Both = 12
Start with the intersection.
Art only = 30 − 12 = 18; music only = 34 − 12 = 22
Each set total already includes those in both.
Neither = 80 − (18+12+22) = 28
The four disjoint region counts now sum to 80.
02 · Wrong outer count
A learner writes 80−30−34 = 16 for neither. Explain.
Hint
The pupils in both were subtracted twice.
Worked solution
Add the overlap back: neither = 80−30−34+12 = 28. Subtracting the disjoint union 52 once is safer.
03 / Translate before calculating
Write which regions satisfy the words.
03 · Both
Find P(A ∩ M).
Hint
Only the intersection qualifies.
Worked solution
12/80 = 3/20.
04 · Art only
Find P(A ∩ M′).
Hint
Inside A but outside M.
Worked solution
18/80 = 9/40.
05 · Not art
Find P(A′).
Hint
Include music only and neither.
Worked solution
(22+28)/80 = 50/80 = 5/8. Equivalently 1−30/80.
04 / Correct the double count
Inclusive union counts the overlap once.
P(A ∪ M) = P(A) + P(M) − P(A ∩ M).
Adding the two set totals counts the overlap twice; subtract one copy.
06 · At least one
Find the probability a pupil studies at least one of art or music.
Hint
At least one is the inclusive union.
Worked solution
(30+34−12)/80 = 52/80 = 13/20.
07 · Exactly one
Find the probability a pupil studies exactly one of the subjects.
Hint
Exclude the overlap completely.
Worked solution
(18+22)/80 = 40/80 = 1/2. Equivalently (30+34−2×12)/80.
05 / Outside a union versus outside an intersection
Neither and not-both describe different events.
08 · Neither
Find P((A ∪ M)′).
Hint
Outside both circles.
Worked solution
28/80 = 7/20. This is P(A′ ∩ M′).
09 · Not both
Find P((A ∩ M)′).
Hint
Every region except the overlap qualifies.
Worked solution
(18+22+28)/80 = 68/80 = 17/20. This is different from neither.
10 · At least one subject not studied
Explain why A′ ∪ M′ is the not-both event.
Hint
Someone is excluded only if neither complement applies.
Worked solution
Every pupil who does not study both must lack art or lack music (or both). Thus A′ ∪ M′ = (A ∩ M)′, giving 17/20.
06 / Work backwards from the union
The addition rule can find missing overlap.
For a different population, P(B)=0.55, P(C)=0.40 and P(B ∪ C)=0.75.Worked example
P(B ∩ C) = 0.55 + 0.40 − 0.75 = 0.20
Rearrange the addition rule.
B only = 0.35; C only = 0.20; neither = 0.25
Every region is nonnegative and the four sum to 1.
11 · Another diagram
Given P(B)=0.6, P(C)=0.5 and P(neither)=0.2, find the overlap.
Hint
The union has probability 0.8.
Worked solution
P(B ∩ C)=0.6+0.5−0.8=0.3. The remaining regions are B only0.3, C only0.2, neither0.2.
07 / Check whether the diagram is possible
Region probabilities cannot be negative.
12 · Impossible overlap
Can P(B)=0.3 and P(B ∩ C)=0.4 hold?
Hint
The intersection is contained in B.
Worked solution
No. It would give B only = 0.3−0.4 = −0.1. An intersection cannot have greater probability than either set.
13 · Too little overlap
If P(B)=0.8 and P(C)=0.7, why must their overlap be at least 0.5?
Hint
The union cannot exceed 1.
Worked solution
0.8+0.7−P(B ∩ C) ≤ 1 implies P(B ∩ C) ≥ 0.5.
14 · No independence assumption
Do these Venn calculations require A and M to be independent?
Hint
Which rule was used?
Worked solution
No. Counting disjoint regions and the addition rule work for dependent as well as independent events. Independence would be an additional property to check, not an assumption to insert.
08 / Let the regions do the counting
Overlap once for a union; overlap zero times for exactly one.
Fill the intersection first, subtract it from each set total and find the outside region last. Translate event notation into non-overlapping regions. Check that every count is nonnegative and the four regions sum to the whole population.
Section 1 of 8 · Separate four regions