Venn Diagrams and Two-way Tables

A Venn diagram sorts things into overlapping groups. Thirty students: 18 play soccer, 14 play tennis, and 7 play both. The order you fill it in matters enormously — always start with the overlap. Seven go in the middle. Then soccer only is 18 minus 7, which is 11, and tennis only is 14 minus 7, which is 7. Finally, neither is 30 minus those three, so 30 minus 25, which is 5. Check: 11 plus 7 plus 7 plus 5 is 30 ✓. The big trap is writing 18 in the soccer circle, because that 18 already includes the seven who play both. Doing that counts those students twice and pushes your total over 30.

Once the Venn is filled in, probabilities come straight off it. The probability a random student plays soccer is the whole soccer circle, 11 plus 7, so 18 over 30, which simplifies to three fifths. Soccer only is just the left crescent, 11 over 30. Both is the overlap, 7 over 30. Neither is outside both circles, 5 over 30, or one sixth. And soccer or tennis is everything inside either circle, 25 over 30, which is five sixths. Watch the wording carefully: 'soccer' means the whole circle including the overlap, while 'soccer only' excludes it. And the denominator is always the total of 30, never the size of one circle.

The same information fits into a two-way table, with one category down the side and the other across the top. Soccer and tennis is 7, soccer without tennis is 11, tennis without soccer is 7, and neither is 5. Then the totals: 18 play soccer, 12 do not, 14 play tennis, 16 do not, and everything comes to 30. The table has one real advantage over the Venn — the totals check your work automatically. Every row must add across to its total and every column must add down to its total. If any of those fails, you have made an error and you know it immediately.

Exams usually give you a partly complete table to finish. The method is to find any row or column with just one gap, fill it, and look again — each new number opens up another. Here the Year 7 row has 24 and a total of 40, so the gap is 16. The bus column has 24 and a total of 45, so Year 8 bus is 21. Now the Year 8 row has 21 and 19, giving a total of 40. And the no-bus column has 16 and 19, giving 35. Finish with the check: 45 plus 35 is 80 ✓, and 40 plus 40 is 80 ✓. Never try to find the hardest cell first; work through the easy gaps and the table completes itself.

There is a formula behind all of this. To count students who play soccer or tennis, you might add 18 and 14 — but that gives 32, more than the whole class of 30. The reason is that the seven who play both have been counted twice, once in each total. So you subtract them: the number who play A or B equals the number in A, plus the number in B, minus the number in both. That is 18 plus 14 minus 7, which is 25 — matching what the Venn showed. The check is built in: if your 'or' count exceeds the total group, you have double-counted the overlap.