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Probability · Venn diagrams & set notation

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Venn diagrams

Building a Venn diagram from worded information — why you fill the overlap first and work outwards, what the number in a region actually counts, the universal set and the "neither" region outside the circles, and how the grand total pins down whichever region is missing.

Probability · Venn diagrams & set notation

Venn diagrams

Building a Venn diagram from worded information — why you fill the overlap first and work outwards, what the number in a region actually counts, the universal set and the "neither" region outside the circles, and how the grand total pins down whichever region is missing.

Why it works

A Venn diagram is a sorting machine. You draw a rectangle for everybody in the survey — the universal set, labelled ξ\xi — and inside it a circle for each group. Every single person then lands in exactly one region of the picture: inside both circles, inside just one of them, or inside none at all. Nobody is left out and nobody appears twice. That "exactly one region each" property is the whole reason the diagram is trustworthy, and it is also the thing beginners break within about ten seconds.

The number in a region counts that region ONLY. Not the circle it sits in — the region. This is the crucial sentence, because a question never hands you region counts. It hands you circle totals: "35 play tennis" is everybody inside the tennis circle, both the tennis-only people and the ones who also play hockey. So the information you are given and the information the diagram wants are two different things, and converting one into the other is the entire skill.

The classic wreck, with numbers. In a group of 6060 students, 3535 play tennis, 2828 play hockey and 1212 play both. Copy those numbers straight onto the diagram — 3535 in the tennis-only part, 1212 in the middle, 2828 in the hockey-only part — and the picture is already dead:

35+12+28=75>60.35 + 12 + 28 = 75 > 60.

Seventy-five people in a group of sixty. And look at what the diagram now claims: the tennis circle holds 35+12=4735 + 12 = 47, so it says 4747 students play tennis when the question said 3535. The 1212 overlap students have been written down twice — once in their own region and once inside the 3535 they were always part of.

So fill the middle first. Do it in this order and the trap cannot spring:
  • the overlap is the only region a question usually gives you directly: both =12= 12,
  • tennis only =35−12=23= 35 - 12 = 23 (take the overlap out of the tennis total),
  • hockey only =28−12=16= 28 - 12 = 16,
  • neither =60−(23+12+16)=60−51=9= 60 - (23 + 12 + 16) = 60 - 51 = 9.
ξ9TH231216Check it: the tennis circle totals 23+12=3523 + 12 = 35 ✓, the hockey circle totals 16+12=2816 + 12 = 28 ✓, and all four regions total 23+12+16+9=6023 + 12 + 16 + 9 = 60 ✓. Every region is a whole number and none is negative — if either of those fails, you have misread the question or subtracted the wrong way round.

The reason the overlap goes first is not tradition, it is dependency. Every other region is defined as a total minus the overlap, so until the overlap is on the page you cannot work out a single other region. Start anywhere else and you are guessing.

The outside region is real. Everything inside the rectangle but outside every circle is the people who are in none of the groups — the answer to "how many play neither?". You cannot see it in the given numbers, so you get it from the grand total:

outside=total−(everything inside the circles).\text{outside} = \text{total} - (\text{everything inside the circles}).

Here 60−51=960 - 51 = 9. The tempting shortcut is to subtract the two given totals: 60−35−28=−360 - 35 - 28 = -3. A negative number of people should stop you dead. It goes wrong because the 1212 both-players sit inside the 3535 and inside the 2828, so that calculation removes them twice. Only 23+12+16=5123 + 12 + 16 = 51 different students play at least one sport, which is why 99, not −3-3, is the answer.

Three circles: same idea, one layer deeper. With three groups the picture has eight regions — three "only" regions, three pairwise slivers, the centre, and the outside — and the same dependency rule applies, so you work from the centre outwards:
  1. the centre (in all three) goes in first;
  2. each pairwise region: take the centre out of the given pair total. If 99
students play football and netball and 44 play all three, then those 44 are already football-and-netball players, so the football-and-netball-only region is 9−4=59 - 4 = 5;
  1. each single region: take everything already inside that circle off its
total. If the football circle must total 2020 and it already holds 55, 22 and 44 in its overlapping regions, then football only =20−(5+2+4)=9= 20 - (5 + 2 + 4) = 9;
  1. the outside: grand total minus the seven regions inside.
Using the total to find a missing region. Sometimes the question withholds the overlap and gives you the outside instead. Then you run the machine in reverse: call the unknown region xx, write every other region in terms of xx, and use the one fact that never changes — *all the regions add up to the grand total*. That single equation pins xx down, because the diagram accounts for every person exactly once.

Two habits make this topic almost mechanical. Write the overlap first, always. Then, before answering anything, add your regions up and check they hit the grand total — a diagram that sums correctly and reproduces every circle total is a diagram you can read answers off with confidence.