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This page sets out the method for Venn diagrams in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Venn diagrams topic page.
When you are told how many are in each set and how many are in both.
When you know the set sizes and the total (or the 'neither' number) but not the overlap.
only A = 2, both = 1, only B = 3, neither = 4. A or B is 3/5. Add the three inside regions and divide by the total.
A and B is the overlap only. only A = 2, both = 1, only B = 3, neither = 4. A and B is 1/10.
Neither is the number outside both circles, over the total.
The probability of A includes only A and the overlap. It does not include only B.
Add only A, both, only B and neither before you write a probability.
Only A is in A but not B. Only B is in B but not A. Both is the lens in the middle.
Label the four numbers before you calculate. Suppose only A is 4, both is 2, only B is 5 and neither is 9. The total is 4 + 2 + 5 + 9 = 20. A probability uses 20 as the denominator, not 4 + 2 + 5, because the people outside the circles are still in the group. Leaving them out makes every probability too big.
Each question names a different collection of regions. A and B is the 2 in the overlap, so the probability is 2/20 = 1/10. Neither is 9/20. Only A is 4/20 = 1/5. Read the words carefully. And means the lens in the middle. Only means a crescent. Neither means outside both circles.
Method: 1. Write down only A, both, only B and neither. 2. Add all four numbers to get the total. 3. Add just the regions the question names. 4. Divide that count by the total and simplify.
Questions often give the size of each set and the size of the overlap, not the four regions ready-made. Put the overlap in first. If 18 students play football, 14 play tennis and 8 play both, the overlap is 8. Only football is 18 - 8 = 10. Only tennis is 14 - 8 = 6. You subtract the overlap from each set so that the 8 are not standing in the crescent as well as in the middle.
The number inside either circle is 10 + 8 + 6 = 24. If the whole group has 30 students, neither is 30 - 24 = 6. The probability of neither is 6/30 = 1/5. If you had put 18 in the only-football region, the football circle would have read as 26, which is more footballers than the question gave. The overlap is part of the 18, not an extra 8.
Method: 1. Write the intersection in the overlap. 2. Only A is the set A total minus the overlap. 3. Only B is the set B total minus the overlap. 4. Neither is the group total minus the three inside regions.
A or B is everything in the circles. From only A = 6, both = 3, only B = 1 and neither = 5, the inside count is 6 + 3 + 1 = 10 and the total is 15. The probability is 10/15 = 2/3. Leaving out the overlap gives 7 out of 15, which misses the people who do both. Or in this sense includes the people who do both.
The same picture gives the subtraction rule. P(A) = 9/15 because A is 6 + 3. P(B) = 4/15. P(both) = 3/15. Then P(A) + P(B) - P(both) = 9/15 + 4/15 - 3/15 = 10/15. You subtract once because the overlap was inside both circles. And is only that subtracted piece. Neither stays outside the calculation for or.
Method: 1. For and, use only the overlap. 2. For or, add only A, both and only B. 3. Or use P(A) + P(B) - P(both). 4. For neither, use the outside number over the total.
A number in the intersection is in set A and in set B. It is one group of people, not two. When you are asked for the probability of A, add only A and the overlap. When you are asked for B, add only B and the same overlap. When you are asked for A or B, add the overlap once. When you are asked for A and B, use the overlap alone. The same 5 can appear in several answers without being five extra people.
Suppose only A is 7, both is 5, only B is 3 and neither is 5. The probability of A is (7 + 5)/20 = 12/20 = 3/5. Using only the crescent, 7/20, drops the people in the overlap, and those people do belong to A. Using (7 + 5 + 3)/20 includes only B, which is not part of A. The whole circle, and nothing outside it, is the set.
Method: 1. Treat the overlap as members of A. 2. Treat the same overlap as members of B. 3. Add it once for A or B. 4. Use it alone for A and B.
Exam Venn diagrams come with a short key: xi is the whole group in the box, and letters such as A and B name the circles. Then the questions use symbols. A cap B means in both circles, which is the overlap. A cup B means in at least one circle, so it is the overlap and both crescents. A' means not in A, which is everything outside circle A, including the neither region.
Command words matter too. "Write down" and "How many" want a single count read straight off the diagram. "Work out the probability" wants a fraction with the total of the whole box underneath. A fraction such as 12/40 usually needs to be given in its simplest form, so look for a common factor before you write the final answer.
Method: 1. Translate the symbol into words, for example A cap B is "in both". 2. Colour or tick the matching regions on the diagram. 3. Add the numbers in those regions only. 4. Put that sum over the total of the whole box, then simplify.
Tip: For A' you can add the regions outside circle A, or work out 1 - P(A). Doing both is a fast check.
Many questions give a story rather than a diagram: "In a group of 60 students, 25 play hockey, 18 play netball and 7 play both." Always write the overlap first. The 25 for hockey is the whole hockey circle, so only hockey is 25 - 7 = 18. Do the same for netball, then add the three regions you know and subtract from the total to find neither.
Sometimes a fraction or percentage is hidden in the wording, such as "3/8 of the people have neither". Turn it into a number first with 3/8 × 160 = 60. When both circles are known as whole totals and neither is known, use the rule: total of the circles minus the overlap equals the number in at least one circle.
Method: 1. Write down the total and any region given directly, such as both or neither. 2. Turn fractions or percentages of the total into whole numbers. 3. Subtract the overlap from each circle total to find the two "only" regions. 4. Check that all four regions add to the total in the box.
Tip: The final check, four regions adding to the total, is worth doing every time. It catches a missed overlap at once.
In another layout, the sets are lists: xi = \1, 2, 3, dots, 12\, A = even numbers, B = factors of 12. First write each set out in full: A = \2, 4, 6, 8, 10, 12\ and B = \1, 2, 3, 4, 6, 12\. A number that appears in both lists goes in the overlap. Every number in xi must be placed exactly once, including the ones that fit neither set.
Once the diagram is complete, count each region as a number. The overlap here is \2, 4, 6, 12\, so it holds 4 numbers. A probability asks "how many numbers", not "which numbers", so P(A cup B) is the count of numbers in at least one circle over the 12 numbers in xi. Do not put a number in the overlap and again in a crescent.
Method: 1. List xi, then write every set out in full. 2. Find the numbers common to both lists and place them in the overlap. 3. Place the remaining members of each set in their own crescent, then the leftovers outside. 4. Count the numbers in each region and use those counts for probabilities.
Some exam questions give you a finished diagram and ask you to criticise it, or ask you to "show that" a count or probability is correct. For a criticism, add the four regions and compare with the stated total, or check that a circle total has not been written in a crescent. For "show that", you must work towards the given answer and write each step, ending with the value you were asked to reach.
The OR rule is P(A cup B) = P(A) + P(B) - P(A cap B). It is useful when the question gives probabilities as decimals rather than counts, and there is no diagram to read. A test for independence uses a different product: events are independent only if P(A cap B) = P(A) × P(B). If the two sides are not equal, say so and quote both values.
Method: 1. For a criticism, add all regions and compare with the stated total. 2. For "show that", write every step and finish on the number you were given. 3. For probabilities without counts, use P(A cup B) = P(A) + P(B) - P(A cap B). 4. For independence, work out both sides of P(A cap B) = P(A) × P(B) and compare.
Tip: In a "show that" question, the final line must match the printed answer exactly. A different answer means you have made a slip somewhere above.
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