GCSE Maths: Sets and Venn diagrams (Higher)
Part of the GCSE Maths probability chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Sets and Venn diagrams
A set is a collection of items called elements. We write a set in curly brackets: A = {2, 4, 6, 8}. The symbol ∈ means “is an element of”, so 4 ∈ A, and ∉ means “is not an element of”, so 5 ∉ A. The universal set, written ξ, is everything we are talking about, for example all the whole numbers from 1 to 10. The empty set {} or ∅ has no elements. n(A) means the number of elements of A, so n(A) = 4 above. If every element of B is also in A we write B ⊂ A (B is a subset of A).
Combining sets. Let A = {even numbers} and B = {multiples of 3} for ξ = {1, 2, ..., 12}, so A = {2, 4, 6, 8, 10, 12} and B = {3, 6, 9, 12}. A ∩ B (“A intersection B”) contains the elements in both A and B: {6, 12}. A ∪ B (“A union B”) contains the elements in A or B or both: {2, 3, 4, 6, 8, 9, 10, 12}. A′ (“A complement” or “not A”) contains everything in ξ that is not in A: {1, 3, 5, 7, 9, 11}. A handy way to remember: ∩ looks like the letter A for “And”, and ∪ is like a cup for “or”.
Venn diagrams. A Venn diagram shows sets as circles inside a rectangle that stands for ξ. The overlap of two circles is A ∩ B. Anything inside the rectangle but outside every circle is in none of the sets. We can write either the elements or the number of elements in each region. When counting, fill in the middle first and then work outwards. Be careful: a number given for a circle (such as “18 play football”) includes the overlap.
Probability from a Venn diagram. If one item is chosen at random from everything in ξ, then all the items are equally likely, so P(event) = n(event) ÷ n(ξ). For example, with 30 students in total and 13 of them in A only, P(A only) = 13/30. The total on the bottom is everything in ξ, including the numbers outside the circles.
Higher: the addition formula. The numbers in A and in B both count the overlap, so P(A) + P(B) counts the overlap twice. Subtract it once: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Also P(A′) = 1 − P(A), and P(neither) = 1 − P(A ∪ B).
Key ideas
- ξ is the universal set. ∈ means “is in”. n(A) is the number of elements in A.
- A ∩ B = in both A and B. A ∪ B = in A or B or both. A′ = not in A.
- Fill in a Venn diagram from the middle outwards. Subtract the overlap from each circle total.
- P(event) = n(event) ÷ n(ξ). The bottom number is everything in ξ.
- Higher: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Write out both sets, then use the meaning of ∩ (both), ∪ (either) and ′ (not in) to pick out the elements.
- Write out the sets. List every element.ξ = {1, 2, ..., 12}
A = {2, 4, 6, 8, 10, 12}
B = {3, 6, 9, 12} - A ∩ B: in both sets. Find the numbers that appear in both lists.A ∩ B = {6, 12}
- A ∪ B: in A or B (or both). Combine the lists, writing each number once.A ∪ B = {2, 3, 4, 6, 8, 9, 10, 12}
- A′: not in A. Everything in ξ that is not even, so the odd numbers.A′ = {1, 3, 5, 7, 9, 11}
Check. n(A) + n(A′) = 6 + 6 = 12 = n(ξ), as it should. ✓
Watch out. Do not write a repeated element twice in A ∪ B.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Fill in the overlap first, then work outwards: subtract the overlap from each circle total, and finally find the number outside both circles from the total.
- Start with the overlap. The people in both sets go in the middle where the circles overlap. This is F ∩ S.Both football and swimming: 5
- Work out F only. The whole of circle F has 18 people, but 5 of them are also in S. They are already in the middle, so take them away.18 − 5 = 13
- Work out S only. In the same way, circle S has 12 people, 5 of whom are in the middle.12 − 5 = 7
- Work out the number in neither set. The numbers in the circles add up to 13 + 5 + 7 = 25. The rest of the 30 are outside both circles.30 − 25 = 5
- Draw the Venn diagram. Write each number in its region. The numbers must add up to the total.
Check. Total of all regions: 13 + 5 + 7 + 5 = 30. Circle F: 13 + 5 = 18. Circle S: 7 + 5 = 12. ✓
Watch out. The number given for a circle includes the overlap, so you must subtract the overlap to find “only”.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Work out which regions belong to each set, then add the numbers in those regions.
- Find the regions of A ∪ B. ∪ means “or”: the items in either set, or both.in A only: 8
in B only: 7
in both A and B: 5 - Add them. Count each region once.8 + 5 + 7 = 20, so n(A ∪ B) = 20
- Find the regions of A′. The dash ′ means “not in”: everything outside that set, including outside the circles.in none of the sets (outside the circles): 4
in B only: 7 - Add them. A′ is everything outside circle A.7 + 4 = 11, so n(A′) = 11
Check. n(A) = 13, and n(A) + n(A′) = 13 + 11 = 24 = n(ξ). ✓
Watch out. A′ includes the number outside both circles.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of items. Add the numbers in every region, including the ones outside the circles. This goes on the bottom.4 + 8 + 7 + 5 = 24
- Find the number of items in the event. ∩ means “and”: only the items in both sets. The dash ′ means “not in”: everything outside that set, including outside the circles.in B only: 7
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(dog but not cat) = 724
Check. The other 17 outcomes do not fit the event, so P(the event does not happen) = 1724. Then 724 + 1724 = 2424 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The total on the bottom is everything in the universal set ξ, not just the numbers inside the circles.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. With three sets, work from the middle outwards: the triple overlap first, then the pairs, then the single circles, and finally the outside.
- Start in the very middle. The people in all three sets go where all three circles overlap.A ∩ B ∩ C: 3
- Work out the regions where exactly two circles overlap. Each pair total includes the middle region, so subtract the middle number.A and B only: 8 − 3 = 5
A and C only: 6 − 3 = 3
B and C only: 5 − 3 = 2 - Work out the regions that are in one circle only. Each circle total includes the middle and the two-way overlaps, so subtract all of them.A only: 20 − 5 − 3 − 3 = 9
B only: 18 − 5 − 2 − 3 = 8
C only: 14 − 3 − 2 − 3 = 6 - Work out the number in none of the sets. Add all seven numbers inside the circles and subtract from the total.9 + 8 + 6 + 5 + 3 + 2 + 3 = 36, so 40 − 36 = 4
- Draw the Venn diagram. Write each number in its region.
Check. All the regions add up to 40, the total. Circle A: 9 + 5 + 3 + 3 = 20. ✓
Watch out. Always start from the middle. Subtracting in the wrong order gives negative numbers.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Use the addition formula P(A ∪ B) = P(A) + P(B) − P(A ∩ B), then use the opposite event for “neither”.
- Write the formula. Adding P(A) and P(B) counts the overlap twice, so subtract it once.P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- Put in the numbers. Add first, then subtract.0.5 + 0.4 − 0.2 = 0.9 − 0.2 = 0.7
- Find “neither”. Neither is the opposite of A or B (or both).P(neither) = 1 − 0.7 = 0.3
Check. A only is 0.5 − 0.2 = 0.3, B only is 0.4 − 0.2 = 0.2, both is 0.2, neither is 0.3. Total: 0.3 + 0.2 + 0.2 + 0.3 = 1. ✓
Watch out. If you forget to subtract P(A ∩ B) your total can exceed what is possible.
Watch out
In a Venn diagram the number given for a circle includes the overlap. If 18 play football and 5 play football and swim, only 13 play football only. Also remember that A′ includes the people outside both circles, and for a probability the bottom number is everyone in ξ.Exercise 23I Sets and Venn diagrams
Non-calculatorMethod. Write out the numbers that are odd.
- Check each number from 1 to 12. Odd numbers are not divisible by 2.1, 3, 5, 7, 9, 11
- Write the set. Use curly brackets.A = {1, 3, 5, 7, 9, 11}
Check. There are 6 odd numbers and 6 even numbers, 12 in all. ✓
Watch out. Use curly brackets for sets.
Method. Write out the prime numbers up to 12.
- Recall what a prime number is. A prime has exactly two factors, 1 and itself. 1 is not prime.Prime numbers: 2, 3, 5, 7, 11
- Write the set. Use curly brackets.B = {2, 3, 5, 7, 11}
Check. 4, 6, 8, 9, 10, 12 each have more than two factors. ✓
Watch out. 1 is not a prime number.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Write out each set, then use the meaning of the symbols to pick out the right elements.
- Write out the elements of each set. We need to see exactly which numbers are in each set.ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
A = {1, 3, 5, 7, 9, 11} and B = {2, 3, 5, 7, 11} - Decide what the symbols mean. ∩ means “and”: only the items in both sets.A ∩ B
- Pick out the elements that fit. Check each number of ξ in turn.A ∩ B = {3, 5, 7, 11}
Check. The number of elements is 4. We can check by counting: n(A ∩ B) = 4. ✓
Watch out. ∩ needs the element to be in both sets. ∪ needs it in at least one. A′ is everything in ξ that is not in A.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Write out each set, then use the meaning of the symbols to pick out the right elements.
- Write out the elements of each set. We need to see exactly which numbers are in each set.ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
A = {1, 3, 5, 7, 9, 11} and B = {2, 3, 5, 7, 11} - Decide what the symbols mean. ∪ means “or”: the items in either set, or both.A ∪ B
- Pick out the elements that fit. Check each number of ξ in turn.A ∪ B = {1, 2, 3, 5, 7, 9, 11}
Check. The number of elements is 7. We can check by counting: n(A ∪ B) = 7. ✓
Watch out. ∩ needs the element to be in both sets. ∪ needs it in at least one. A′ is everything in ξ that is not in A.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Write out each set, then use the meaning of the symbols to pick out the right elements.
- Write out the elements of each set. We need to see exactly which numbers are in each set.ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
A = {1, 3, 5, 7, 9, 11} and B = {2, 3, 5, 7, 11} - Decide what the symbols mean. The dash ′ means “not in”: everything outside that set, including outside the circles.A′
- Pick out the elements that fit. Check each number of ξ in turn.A′ = {2, 4, 6, 8, 10, 12}
Check. The number of elements is 6. We can check by counting: n(A′) = 6. ✓
Watch out. ∩ needs the element to be in both sets. ∪ needs it in at least one. A′ is everything in ξ that is not in A.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Write out each set, then use the meaning of the symbols to pick out the right elements.
- Write out the elements of each set. We need to see exactly which numbers are in each set.ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
A = {1, 3, 5, 7, 9, 11} and B = {2, 3, 5, 7, 11} - Decide what the symbols mean. ∪ means “or”: the items in either set, or both. The dash ′ means “not in”: everything outside that set, including outside the circles.(A ∪ B)′
- Pick out the elements that fit. Check each number of ξ in turn.(A ∪ B)′ = {4, 6, 8, 10, 12}
Check. The number of elements is 5. We can check by counting: n((A ∪ B)′) = 5. ✓
Watch out. ∩ needs the element to be in both sets. ∪ needs it in at least one. A′ is everything in ξ that is not in A.
Method. n(A) means the number of elements in A.
- Count the elements of A. Count each one once.1, 2, 3, 4, 5, 6: that is 6 elements
- Write the answer. n(A) = 6
Check. 6 numbers are listed, so n(A) = 6. ✓
Watch out. n(A) is a number, not a list.
Method. First find A ∩ B, the elements in both sets, then count them.
- List A ∩ B. Numbers in both lists.A ∩ B = {4, 5, 6}
- Count them. n(A ∩ B) = 3
Check. 4, 5, 6 appear in both lists. ✓
Watch out. Do not count numbers that are in only one of the sets.
Method. Find A ∪ B, the elements in either set, then count them.
- List A ∪ B. Write each element once.A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8}
- Count them. n(A ∪ B) = 8
Check. n(A) + n(B) − n(A ∩ B) = 6 + 5 − 3 = 8. ✓
Watch out. Overlap elements are only counted once.
Method. Find the elements of ξ that are in neither A nor B.
- List A ∪ B. From the previous part.A ∪ B = {1, 2, ..., 8}
- Find what is left in ξ. ξ = {1, ..., 9}, so only 9 is not in A ∪ B.(A ∪ B)′ = {9}
- Count. n((A ∪ B)′) = 1
Check. n(A ∪ B) + n((A ∪ B)′) = 8 + 1 = 9 = n(ξ). ✓
Watch out. Do not forget to look at ξ to find the missing element.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Fill in the overlap first, then work outwards: subtract the overlap from each circle total, and finally find the number outside both circles from the total.
- Start with the overlap. The people in both sets go in the middle where the circles overlap. This is A ∩ B.Both football and swimming: 5
- Work out A only. The whole of circle A has 18 people, but 5 of them are also in B. They are already in the middle, so take them away.18 − 5 = 13
- Work out B only. In the same way, circle B has 12 people, 5 of whom are in the middle.12 − 5 = 7
- Work out the number in neither set. The numbers in the circles add up to 13 + 5 + 7 = 25. The rest of the 30 are outside both circles.30 − 25 = 5
- Draw the Venn diagram. Write each number in its region. The numbers must add up to the total.
Check. Total of all regions: 13 + 5 + 7 + 5 = 30. Circle A: 13 + 5 = 18. Circle B: 7 + 5 = 12. ✓
Watch out. The number given for a circle includes the overlap, so you must subtract the overlap to find “only”.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Fill in the overlap first, then work outwards: subtract the overlap from each circle total, and finally find the number outside both circles from the total.
- Start with the overlap. The people in both sets go in the middle where the circles overlap. This is A ∩ B.Both cat and dog: 6
- Work out A only. The whole of circle A has 25 people, but 6 of them are also in B. They are already in the middle, so take them away.25 − 6 = 19
- Work out B only. In the same way, circle B has 17 people, 6 of whom are in the middle.17 − 6 = 11
- Work out the number in neither set. The numbers in the circles add up to 19 + 6 + 11 = 36. The rest of the 40 are outside both circles.40 − 36 = 4
- Draw the Venn diagram. Write each number in its region. The numbers must add up to the total.
Check. Total of all regions: 19 + 6 + 11 + 4 = 40. Circle A: 19 + 6 = 25. Circle B: 11 + 6 = 17. ✓
Watch out. The number given for a circle includes the overlap, so you must subtract the overlap to find “only”.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Fill in the overlap first, then work outwards: subtract the overlap from each circle total, and finally find the number outside both circles from the total.
- Start with the overlap. The people in both sets go in the middle where the circles overlap. This is A ∩ B.Both French and Spanish: 10
- Work out A only. The whole of circle A has 28 people, but 10 of them are also in B. They are already in the middle, so take them away.28 − 10 = 18
- Work out B only. In the same way, circle B has 25 people, 10 of whom are in the middle.25 − 10 = 15
- Work out the number in neither set. The numbers in the circles add up to 18 + 10 + 15 = 43. The rest of the 50 are outside both circles.50 − 43 = 7
- Draw the Venn diagram. Write each number in its region. The numbers must add up to the total.
Check. Total of all regions: 18 + 10 + 15 + 7 = 50. Circle A: 18 + 10 = 28. Circle B: 15 + 10 = 25. ✓
Watch out. The number given for a circle includes the overlap, so you must subtract the overlap to find “only”.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Find the regions of the Venn diagram that make up the set, then add the numbers inside them.
- Decide which regions of the diagram belong to the set. This is just the set itself, so we take every region inside its circle.in A only: 9
in both A and B: 4 - Add the numbers in those regions. Each region holds a different group of items, so we add them to count the whole set.9 + 4 = 13
Check. All the numbers in the diagram add up to 22. Counting the other regions gives 9, and 13 + 9 = 22. ✓
Watch out. Count each region only once. Numbers in an overlap belong to both circles, so make sure you include them when the question says “A or B”.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Find the regions of the Venn diagram that make up the set, then add the numbers inside them.
- Decide which regions of the diagram belong to the set. ∩ means “and”: only the items in both sets.in both A and B: 4
- Read off the number. There is only one region to count.n(A ∩ B) = 4
Check. All the numbers in the diagram add up to 22. Counting the other regions gives 18, and 4 + 18 = 22. ✓
Watch out. Count each region only once. Numbers in an overlap belong to both circles, so make sure you include them when the question says “A or B”.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Find the regions of the Venn diagram that make up the set, then add the numbers inside them.
- Decide which regions of the diagram belong to the set. ∪ means “or”: the items in either set, or both.in A only: 9
in B only: 6
in both A and B: 4 - Add the numbers in those regions. Each region holds a different group of items, so we add them to count the whole set.9 + 6 + 4 = 19
Check. All the numbers in the diagram add up to 22. Counting the other regions gives 3, and 19 + 3 = 22. ✓
Watch out. Count each region only once. Numbers in an overlap belong to both circles, so make sure you include them when the question says “A or B”.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Find the regions of the Venn diagram that make up the set, then add the numbers inside them.
- Decide which regions of the diagram belong to the set. The dash ′ means “not in”: everything outside that set, including outside the circles.in none of the sets (outside the circles): 3
in B only: 6 - Add the numbers in those regions. Each region holds a different group of items, so we add them to count the whole set.3 + 6 = 9
Check. All the numbers in the diagram add up to 22. Counting the other regions gives 13, and 9 + 13 = 22. ✓
Watch out. Count each region only once. Numbers in an overlap belong to both circles, so make sure you include them when the question says “A or B”.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Find the regions of the Venn diagram that make up the set, then add the numbers inside them.
- Decide which regions of the diagram belong to the set. ∩ means “and”: only the items in both sets. The dash ′ means “not in”: everything outside that set, including outside the circles.in A only: 9
- Read off the number. There is only one region to count.n(A ∩ B′) = 9
Check. All the numbers in the diagram add up to 22. Counting the other regions gives 13, and 9 + 13 = 22. ✓
Watch out. Count each region only once. Numbers in an overlap belong to both circles, so make sure you include them when the question says “A or B”.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Find the regions of the Venn diagram that make up the set, then add the numbers inside them.
- Decide which regions of the diagram belong to the set. ∪ means “or”: the items in either set, or both. The dash ′ means “not in”: everything outside that set, including outside the circles.in none of the sets (outside the circles): 3
- Read off the number. There is only one region to count.n((A ∪ B)′) = 3
Check. All the numbers in the diagram add up to 22. Counting the other regions gives 19, and 3 + 19 = 22. ✓
Watch out. Count each region only once. Numbers in an overlap belong to both circles, so make sure you include them when the question says “A or B”.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Look at which regions are shaded and match them to the symbols ∩ (and), ∪ (or) and ′ (not).
- List the shaded regions. Describe each shaded part in words.in both A and B
- Match them to set notation. ∩ means “and”: only the items in both sets.A ∩ B
- Check by shading. Shade the regions described by your answer and compare with the picture.A ∩ B matches the picture.
Check. The region A ∩ B contains 1 of the regions of the diagram, which agrees with the picture. ✓
Watch out. There can be more than one correct way to describe the same region, for example (A ∪ B)′ = A′ ∩ B′.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Look at which regions are shaded and match them to the symbols ∩ (and), ∪ (or) and ′ (not).
- List the shaded regions. Describe each shaded part in words.in A only
in B only
in both A and B - Match them to set notation. ∪ means “or”: the items in either set, or both.A ∪ B
- Check by shading. Shade the regions described by your answer and compare with the picture.A ∪ B matches the picture.
Check. The region A ∪ B contains 3 of the regions of the diagram, which agrees with the picture. ✓
Watch out. There can be more than one correct way to describe the same region, for example (A ∪ B)′ = A′ ∩ B′.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Look at which regions are shaded and match them to the symbols ∩ (and), ∪ (or) and ′ (not).
- List the shaded regions. Describe each shaded part in words.in none of the sets (outside the circles)
in B only - Match them to set notation. The dash ′ means “not in”: everything outside that set, including outside the circles.A′
- Check by shading. Shade the regions described by your answer and compare with the picture.A′ matches the picture.
Check. The region A′ contains 2 of the regions of the diagram, which agrees with the picture. ✓
Watch out. There can be more than one correct way to describe the same region, for example (A ∪ B)′ = A′ ∩ B′.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Look at which regions are shaded and match them to the symbols ∩ (and), ∪ (or) and ′ (not).
- List the shaded regions. Describe each shaded part in words.in A only
- Match them to set notation. ∩ means “and”: only the items in both sets. The dash ′ means “not in”: everything outside that set, including outside the circles.A ∩ B′
- Check by shading. Shade the regions described by your answer and compare with the picture.A ∩ B′ matches the picture.
Check. The region A ∩ B′ contains 1 of the regions of the diagram, which agrees with the picture. ✓
Watch out. There can be more than one correct way to describe the same region, for example (A ∪ B)′ = A′ ∩ B′.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Look at which regions are shaded and match them to the symbols ∩ (and), ∪ (or) and ′ (not).
- List the shaded regions. Describe each shaded part in words.in none of the sets (outside the circles)
- Match them to set notation. ∪ means “or”: the items in either set, or both. The dash ′ means “not in”: everything outside that set, including outside the circles.(A ∪ B)′
- Check by shading. Shade the regions described by your answer and compare with the picture.(A ∪ B)′ matches the picture.
Check. The region (A ∪ B)′ contains 1 of the regions of the diagram, which agrees with the picture. ✓
Watch out. There can be more than one correct way to describe the same region, for example (A ∪ B)′ = A′ ∩ B′.
Method. ∅ is the empty set, with no elements. Think about what it means for the overlap to be empty.
- Read the statement. A ∩ B = ∅ says there are no elements in both A and B.overlap is empty
- Say what this means for the Venn diagram. The circles do not overlap.The circles are separate
- Give an example. The set of odd numbers and the set of even numbers share nothing.A = {1, 3, 5}, B = {2, 4, 6}: A ∩ B = ∅
Check. This is the same idea as mutually exclusive events in probability. ✓
Watch out. Do not confuse ∅ with 0: the empty set has no elements, and n(∅) = 0.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Fill in the overlap first, then work outwards: subtract the overlap from each circle total, and finally find the number outside both circles from the total.
- Start with the overlap. The people in both sets go in the middle where the circles overlap. This is B ∩ S.Both a brother and a sister: 7
- Work out B only. The whole of circle B has 15 people, but 7 of them are also in S. They are already in the middle, so take them away.15 − 7 = 8
- Work out S only. In the same way, circle S has 12 people, 7 of whom are in the middle.12 − 7 = 5
- Work out the number in neither set. The numbers in the circles add up to 8 + 7 + 5 = 20. The rest of the 28 are outside both circles.28 − 20 = 8
- Draw the Venn diagram. Write each number in its region. The numbers must add up to the total.
Check. Total of all regions: 8 + 7 + 5 + 8 = 28. Circle B: 8 + 7 = 15. Circle S: 5 + 7 = 12. ✓
Watch out. The number given for a circle includes the overlap, so you must subtract the overlap to find “only”.
Method. Add n(A) and n(B), but the overlap has been counted twice, so subtract it once.
- Add the two circle totals. This counts the elements in the overlap twice.14 + 9 = 23
- Subtract the overlap once. Take away the double-counted elements.23 − 4 = 19
- State the result. n(A ∪ B) = 19
Check. A only = 14 − 4 = 10, B only = 9 − 4 = 5, both = 4: 10 + 5 + 4 = 19. ✓
Watch out. Do not just add n(A) and n(B).
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Find the regions of the Venn diagram that make up the set, then add the numbers inside them.
- Decide which regions of the diagram belong to the set. ∩ means “and”: only the items in both sets.in all three sets: 1
- Read off the number. There is only one region to count.n(A ∩ B ∩ C) = 1
Check. All the numbers in the diagram add up to 30. Counting the other regions gives 29, and 1 + 29 = 30. ✓
Watch out. Count each region only once. Numbers in an overlap belong to both circles, so make sure you include them when the question says “A or B”.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Find the regions of the Venn diagram that make up the set, then add the numbers inside them.
- Decide which regions of the diagram belong to the set. ∪ means “or”: the items in either set, or both.in A only: 5
in B only: 4
in A and B only: 3
in C only: 6
in A and C only: 2
in B and C only: 4
in all three sets: 1 - Add the numbers in those regions. Each region holds a different group of items, so we add them to count the whole set.5 + 4 + 3 + 6 + 2 + 4 + 1 = 25
Check. All the numbers in the diagram add up to 30. Counting the other regions gives 5, and 25 + 5 = 30. ✓
Watch out. Count each region only once. Numbers in an overlap belong to both circles, so make sure you include them when the question says “A or B”.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Find the regions of the Venn diagram that make up the set, then add the numbers inside them.
- Decide which regions of the diagram belong to the set. ∩ means “and”: only the items in both sets. The dash ′ means “not in”: everything outside that set, including outside the circles.in A and B only: 3
- Read off the number. There is only one region to count.n(A ∩ B ∩ C′) = 3
Check. All the numbers in the diagram add up to 30. Counting the other regions gives 27, and 3 + 27 = 30. ✓
Watch out. Count each region only once. Numbers in an overlap belong to both circles, so make sure you include them when the question says “A or B”.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Find the regions of the Venn diagram that make up the set, then add the numbers inside them.
- Decide which regions of the diagram belong to the set. ∪ means “or”: the items in either set, or both. The dash ′ means “not in”: everything outside that set, including outside the circles.in none of the sets (outside the circles): 5
- Read off the number. There is only one region to count.n((A ∪ B ∪ C)′) = 5
Check. All the numbers in the diagram add up to 30. Counting the other regions gives 25, and 5 + 25 = 30. ✓
Watch out. Count each region only once. Numbers in an overlap belong to both circles, so make sure you include them when the question says “A or B”.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Find the regions of the Venn diagram that make up the set, then add the numbers inside them.
- Decide which regions of the diagram belong to the set. ∩ means “and”: only the items in both sets. The dash ′ means “not in”: everything outside that set, including outside the circles.in B only: 4
in A and B only: 3 - Add the numbers in those regions. Each region holds a different group of items, so we add them to count the whole set.4 + 3 = 7
Check. All the numbers in the diagram add up to 30. Counting the other regions gives 23, and 7 + 23 = 30. ✓
Watch out. Count each region only once. Numbers in an overlap belong to both circles, so make sure you include them when the question says “A or B”.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. With three sets, work from the middle outwards: the triple overlap first, then the pairs, then the single circles, and finally the outside.
- Start in the very middle. The people in all three sets go where all three circles overlap.A ∩ B ∩ C: 4
- Work out the regions where exactly two circles overlap. Each pair total includes the middle region, so subtract the middle number.A and B only: 12 − 4 = 8
A and C only: 10 − 4 = 6
B and C only: 9 − 4 = 5 - Work out the regions that are in one circle only. Each circle total includes the middle and the two-way overlaps, so subtract all of them.A only: 30 − 8 − 6 − 4 = 12
B only: 28 − 8 − 5 − 4 = 11
C only: 25 − 6 − 5 − 4 = 10 - Work out the number in none of the sets. Add all seven numbers inside the circles and subtract from the total.12 + 11 + 10 + 8 + 6 + 5 + 4 = 56, so 60 − 56 = 4
- Draw the Venn diagram. Write each number in its region.
Check. All the regions add up to 60, the total. Circle A: 12 + 8 + 6 + 4 = 30. ✓
Watch out. Always start from the middle. Subtracting in the wrong order gives negative numbers.
Exercise 23J Probability from Venn diagrams
Non-calculatorKey words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of items. Add the numbers in every region, including the ones outside the circles. This goes on the bottom.4 + 8 + 7 + 5 = 24
- Find the number of items in the event. This is just the set itself, so we take every region inside its circle.in A only: 8
in both A and B: 5
8 + 5 = 13 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the student is in A) = 1324
Check. The other 11 outcomes do not fit the event, so P(the event does not happen) = 1124. Then 1324 + 1124 = 2424 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The total on the bottom is everything in the universal set ξ, not just the numbers inside the circles.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of items. Add the numbers in every region, including the ones outside the circles. This goes on the bottom.4 + 8 + 7 + 5 = 24
- Find the number of items in the event. This is just the set itself, so we take every region inside its circle.in B only: 7
in both A and B: 5
7 + 5 = 12 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the student is in B) = 1224
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 12. The value does not change but the numbers are smaller.12 ÷ 12 = 1 and 24 ÷ 12 = 2
1224 = 12
Check. The other 12 outcomes do not fit the event, so P(the event does not happen) = 12. Then 1224 + 1224 = 2424 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The total on the bottom is everything in the universal set ξ, not just the numbers inside the circles.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of items. Add the numbers in every region, including the ones outside the circles. This goes on the bottom.4 + 8 + 7 + 5 = 24
- Find the number of items in the event. ∩ means “and”: only the items in both sets.in both A and B: 5
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the student is in A ∩ B) = 524
Check. The other 19 outcomes do not fit the event, so P(the event does not happen) = 1924. Then 524 + 1924 = 2424 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The total on the bottom is everything in the universal set ξ, not just the numbers inside the circles.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of items. Add the numbers in every region, including the ones outside the circles. This goes on the bottom.4 + 8 + 7 + 5 = 24
- Find the number of items in the event. ∪ means “or”: the items in either set, or both.in A only: 8
in B only: 7
in both A and B: 5
8 + 7 + 5 = 20 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the student is in A ∪ B) = 2024
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 4. The value does not change but the numbers are smaller.20 ÷ 4 = 5 and 24 ÷ 4 = 6
2024 = 56
Check. The other 4 outcomes do not fit the event, so P(the event does not happen) = 16. Then 2024 + 424 = 2424 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The total on the bottom is everything in the universal set ξ, not just the numbers inside the circles.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of items. Add the numbers in every region, including the ones outside the circles. This goes on the bottom.4 + 8 + 7 + 5 = 24
- Find the number of items in the event. The dash ′ means “not in”: everything outside that set, including outside the circles.in none of the sets (outside the circles): 4
in B only: 7
4 + 7 = 11 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the student is not in A) = 1124
Check. The other 13 outcomes do not fit the event, so P(the event does not happen) = 1324. Then 1124 + 1324 = 2424 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The total on the bottom is everything in the universal set ξ, not just the numbers inside the circles.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of items. Add the numbers in every region, including the ones outside the circles. This goes on the bottom.4 + 8 + 7 + 5 = 24
- Find the number of items in the event. ∩ means “and”: only the items in both sets. The dash ′ means “not in”: everything outside that set, including outside the circles.in A only: 8
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the student is in A only) = 824
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 8. The value does not change but the numbers are smaller.8 ÷ 8 = 1 and 24 ÷ 8 = 3
824 = 13
Check. The other 16 outcomes do not fit the event, so P(the event does not happen) = 23. Then 824 + 1624 = 2424 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The total on the bottom is everything in the universal set ξ, not just the numbers inside the circles.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of items. Add the numbers in every region, including the ones outside the circles. This goes on the bottom.5 + 13 + 7 + 5 = 30
- Find the number of items in the event. ∩ means “and”: only the items in both sets. The dash ′ means “not in”: everything outside that set, including outside the circles.in F only: 13
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(plays football only) = 1330
Check. The other 17 outcomes do not fit the event, so P(the event does not happen) = 1730. Then 1330 + 1730 = 3030 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The total on the bottom is everything in the universal set ξ, not just the numbers inside the circles.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of items. Add the numbers in every region, including the ones outside the circles. This goes on the bottom.5 + 13 + 7 + 5 = 30
- Find the number of items in the event. ∪ means “or”: the items in either set, or both. The dash ′ means “not in”: everything outside that set, including outside the circles.in none of the sets (outside the circles): 5
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(plays neither sport) = 530
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 5. The value does not change but the numbers are smaller.5 ÷ 5 = 1 and 30 ÷ 5 = 6
530 = 16
Check. The other 25 outcomes do not fit the event, so P(the event does not happen) = 56. Then 530 + 2530 = 3030 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The total on the bottom is everything in the universal set ξ, not just the numbers inside the circles.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of items. Add the numbers in every region, including the ones outside the circles. This goes on the bottom.5 + 13 + 7 + 5 = 30
- Find the number of items in the event. ∪ means “or”: the items in either set, or both.in F only: 13
in S only: 7
in both F and S: 5
13 + 7 + 5 = 25 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(plays at least one sport) = 2530
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 5. The value does not change but the numbers are smaller.25 ÷ 5 = 5 and 30 ÷ 5 = 6
2530 = 56
Check. The other 5 outcomes do not fit the event, so P(the event does not happen) = 16. Then 2530 + 530 = 3030 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The total on the bottom is everything in the universal set ξ, not just the numbers inside the circles.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of items. Add the numbers in every region, including the ones outside the circles. This goes on the bottom.5 + 13 + 7 + 5 = 30
- Find the number of items in the event. ∩ means “and”: only the items in both sets. ∪ means “or”: the items in either set, or both. The dash ′ means “not in”: everything outside that set, including outside the circles.in F only: 13
in S only: 7
13 + 7 = 20 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(plays exactly one sport) = 2030
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 10. The value does not change but the numbers are smaller.20 ÷ 10 = 2 and 30 ÷ 10 = 3
2030 = 23
Check. The other 10 outcomes do not fit the event, so P(the event does not happen) = 13. Then 2030 + 1030 = 3030 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The total on the bottom is everything in the universal set ξ, not just the numbers inside the circles.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of items. Add the numbers in every region, including the ones outside the circles. This goes on the bottom.4 + 19 + 11 + 6 = 40
- Find the number of items in the event. ∩ means “and”: only the items in both sets. The dash ′ means “not in”: everything outside that set, including outside the circles.in C only: 19
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(owns a cat only) = 1940
Check. The other 21 outcomes do not fit the event, so P(the event does not happen) = 2140. Then 1940 + 2140 = 4040 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The total on the bottom is everything in the universal set ξ, not just the numbers inside the circles.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of items. Add the numbers in every region, including the ones outside the circles. This goes on the bottom.4 + 19 + 11 + 6 = 40
- Find the number of items in the event. This is just the set itself, so we take every region inside its circle.in D only: 11
in both C and D: 6
11 + 6 = 17 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(owns a dog) = 1740
Check. The other 23 outcomes do not fit the event, so P(the event does not happen) = 2340. Then 1740 + 2340 = 4040 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The total on the bottom is everything in the universal set ξ, not just the numbers inside the circles.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of items. Add the numbers in every region, including the ones outside the circles. This goes on the bottom.4 + 19 + 11 + 6 = 40
- Find the number of items in the event. The dash ′ means “not in”: everything outside that set, including outside the circles.in none of the sets (outside the circles): 4
in D only: 11
4 + 11 = 15 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(owns no cat) = 1540
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 5. The value does not change but the numbers are smaller.15 ÷ 5 = 3 and 40 ÷ 5 = 8
1540 = 38
Check. The other 25 outcomes do not fit the event, so P(the event does not happen) = 58. Then 1540 + 2540 = 4040 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The total on the bottom is everything in the universal set ξ, not just the numbers inside the circles.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of items. Add the numbers in every region, including the ones outside the circles. This goes on the bottom.4 + 19 + 11 + 6 = 40
- Find the number of items in the event. ∪ means “or”: the items in either set, or both. The dash ′ means “not in”: everything outside that set, including outside the circles.in none of the sets (outside the circles): 4
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(owns neither pet) = 440
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 4. The value does not change but the numbers are smaller.4 ÷ 4 = 1 and 40 ÷ 4 = 10
440 = 110
Check. The other 36 outcomes do not fit the event, so P(the event does not happen) = 910. Then 440 + 3640 = 4040 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The total on the bottom is everything in the universal set ξ, not just the numbers inside the circles.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of items. Add the numbers in every region, including the ones outside the circles. This goes on the bottom.28 + 20 + 12 + 0 = 60
- Find the number of items in the event. ∪ means “or”: the items in either set, or both.in W only: 20
in C only: 12
in both W and C: 0
20 + 12 + 0 = 32 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(walks or cycles) = 3260
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 4. The value does not change but the numbers are smaller.32 ÷ 4 = 8 and 60 ÷ 4 = 15
3260 = 815
Check. The other 28 outcomes do not fit the event, so P(the event does not happen) = 715. Then 3260 + 2860 = 6060 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The total on the bottom is everything in the universal set ξ, not just the numbers inside the circles.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of items. Add the numbers in every region, including the ones outside the circles. This goes on the bottom.13 + 8 + 5 + 4 = 30
- Find the number of items in the event. ∪ means “or”: the items in either set, or both. The dash ′ means “not in”: everything outside that set, including outside the circles.in none of the sets (outside the circles): 13
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(in neither A nor B) = 1330
Check. The other 17 outcomes do not fit the event, so P(the event does not happen) = 1730. Then 1330 + 1730 = 3030 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The total on the bottom is everything in the universal set ξ, not just the numbers inside the circles.
Method. Check whether Ali’s answer is possible and find his mistake.
- Look at his answer. A probability can never be larger than 1.2524 > 1, which is impossible
- Find the mistake. He added 13 and 12. The 5 students in the overlap are counted in both numbers, so they have been counted twice.The overlap was counted twice
- Do it correctly. Subtract the overlap once, or count the regions of A ∪ B directly: 8 + 5 + 7 = 20.P(A ∪ B) = 2024 = 56
Check. 13 + 12 − 5 = 20 students, and 20/24 = 5/6. ✓
Watch out. A “probability” larger than 1 is always a sign of an error.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of items. Add the numbers in every region, including the ones outside the circles. This goes on the bottom.12 + 7 + 5 + 3 + 6 + 2 + 4 + 1 = 40
- Find the number of items in the event. ∩ means “and”: only the items in both sets.in all three sets: 1
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(uses all three) = 140
Check. The other 39 outcomes do not fit the event, so P(the event does not happen) = 3940. Then 140 + 3940 = 4040 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The total on the bottom is everything in the universal set ξ, not just the numbers inside the circles.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of items. Add the numbers in every region, including the ones outside the circles. This goes on the bottom.12 + 7 + 5 + 3 + 6 + 2 + 4 + 1 = 40
- Find the number of items in the event. ∩ means “and”: only the items in both sets. The dash ′ means “not in”: everything outside that set, including outside the circles.in A and B only: 3
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(uses bus and bike only) = 340
Check. The other 37 outcomes do not fit the event, so P(the event does not happen) = 3740. Then 340 + 3740 = 4040 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The total on the bottom is everything in the universal set ξ, not just the numbers inside the circles.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of items. Add the numbers in every region, including the ones outside the circles. This goes on the bottom.12 + 7 + 5 + 3 + 6 + 2 + 4 + 1 = 40
- Find the number of items in the event. ∩ means “and”: only the items in both sets. ∪ means “or”: the items in either set, or both. The dash ′ means “not in”: everything outside that set, including outside the circles.in A only: 7
in B only: 5
in C only: 6
7 + 5 + 6 = 18 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(uses exactly one type) = 1840
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 2. The value does not change but the numbers are smaller.18 ÷ 2 = 9 and 40 ÷ 2 = 20
1840 = 920
Check. The other 22 outcomes do not fit the event, so P(the event does not happen) = 1120. Then 1840 + 2240 = 4040 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The total on the bottom is everything in the universal set ξ, not just the numbers inside the circles.
Key words. A set is a collection of items. ξ is the universal set (everything we are considering). A ∩ B (“A intersection B”) is what is in both A and B. A ∪ B (“A union B”) is what is in A or B or both. A′ is what is not in A. n(A) is the number of items in A.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of items. Add the numbers in every region, including the ones outside the circles. This goes on the bottom.12 + 7 + 5 + 3 + 6 + 2 + 4 + 1 = 40
- Find the number of items in the event. ∪ means “or”: the items in either set, or both. The dash ′ means “not in”: everything outside that set, including outside the circles.in none of the sets (outside the circles): 12
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(uses none) = 1240
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 4. The value does not change but the numbers are smaller.12 ÷ 4 = 3 and 40 ÷ 4 = 10
1240 = 310
Check. The other 28 outcomes do not fit the event, so P(the event does not happen) = 710. Then 1240 + 2840 = 4040 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The total on the bottom is everything in the universal set ξ, not just the numbers inside the circles.
Method. Use the addition formula.
- Write the formula. The overlap is counted twice when we add, so subtract it once.P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- Substitute. Use the given probabilities.0.6 + 0.3 − 0.1 = 0.9 − 0.1 = 0.8
Check. A only 0.5, both 0.1, B only 0.2: 0.5 + 0.1 + 0.2 = 0.8. ✓
Watch out. Subtract the overlap once.
Method. Neither is the opposite of “A or B”.
- Use the answer from (a). P(A ∪ B) = 0.8.P(A ∪ B) = 0.8
- Subtract from 1. P(neither) = 1 − P(A ∪ B).1 − 0.8 = 0.2
Check. 0.8 + 0.2 = 1. ✓
Watch out. Use the opposite event.
Method. A ∩ B′ is A without the overlap.
- Understand the region. A ∩ B′ means in A but not in B, which is A only.A only = A − overlap
- Subtract. P(A only) = P(A) − P(A ∩ B).0.6 − 0.1 = 0.5
Check. A only + both = 0.5 + 0.1 = 0.6 = P(A). ✓
Watch out. A only is not the same as A.
Method. Rearrange the addition formula to find the overlap, then decide whether it is zero.
- Write the formula. P(A ∪ B) = P(A) + P(B) − P(A ∩ B).0.65 = 0.45 + 0.35 − P(A ∩ B)
- Add P(A) and P(B). Simplify the right-hand side.0.65 = 0.8 − P(A ∩ B)
- Solve for P(A ∩ B). Swap the terms: P(A ∩ B) = 0.8 − 0.65.P(A ∩ B) = 0.15
- Decide. Mutually exclusive means P(A ∩ B) = 0.0.15 ≠ 0, so A and B are not mutually exclusive.
Check. 0.45 + 0.35 − 0.15 = 0.65. ✓
Watch out. If events were mutually exclusive then P(A ∪ B) would equal P(A) + P(B) = 0.8, not 0.65.