S1: Venn Diagrams A Venn diagram is a way to represent sets of people or objects. A Venn diagram consists of a rectangle (to represent the whole set) with circles in it (to represent more interesting sets). Example: Out of a group of 1000 students, 58 got A grades in A-Level Maths and 138 got B grades. Draw a Venn diagram to show this. represents whole set of 1000 people represents A grade students represents B grade students A 58 B 138 Write the remaining people in the top right of the main rectangle. 804

Venn diagrams are useful when it is possible to be in more than one of the smaller sets, i.e. the sets are not mutually exclusive. To show this, you need to make the circles that represent the smaller sets overlap – but the maths of the diagram changes. Example: A restaurant holds 100 people. Ten have fish and chips. Thirty-three have chips and fifty-two have fish. Draw a Venn diagram of this. FishChips How do we work out the right numbers to put in the circles? Start with the overlap. It was 10: people are having fish, but there are already 10 people in that circle. 42 remain in the ‘main’ part: more for chips: left: S1: Venn Diagrams

You may be asked to draw a Venn diagram involving three sets. Example: 3000 people are at a concert. 485 wear glasses (G), 2221 are female (F) and 241 have red hair (R). There are 14 female red-haired glasses wearers. 300 glasses wearers are women, 82 glasses wearers are red-haired. There are 76 red-haired females. Draw a Venn diagram to represent this. Start with the sets. They must all overlap: R G F As before, now the overlap figure (14): 14 Fill in the gaps. 76 red-haired females; 14 already gone, so 62: 62 Similarly, 68 in R and G only section: left in R only: in F and G only: in G only: in F only: left over: 495 S1: Venn Diagrams

Questions are written using the proper notation. Remember: “A ∩ B” means “both A and B happening at the same time” (said “A intersect B”). This is shown on a Venn diagram by the overlap of the circles. AB S1: Venn Diagrams

“A U B” means “either A or B” (said “A union B”). This is shown on a Venn diagram by everything inside the circles. A B S1: Venn Diagrams

“A´ ” means “everything that is not A” (said “A complement”). This is represented on a Venn diagram by everything outside the circle for A. A B S1: Venn Diagrams

Example: 30 students are asked if they have seen two films: Avatar (A) and X-Men: First Class (X). A ∩ X = 5. X = people have not seen either film. Draw a Venn diagram to represent this situation. Start with the sets. They must overlap: AX As before, now the overlap figure (5): 5 Which gap can be filled in next? X totals 17. It has 5 in it already, so 12 must be left: 12 3 people have not seen either film: 3 The diagram must have 30 people in it. There are 20 so far, so the A only section must contain 10: 10 Done! Now the table can be used to answer questions. What is the probability that a student chosen at random has seen Avatar? S1: Venn Diagrams

Once complete, Venn diagrams can be used to find the probability of certain events taking place. Let’s look at the diagrams from this presentation so far. A 58 B What is the probability that a randomly chosen student has an A? There are 58 A students out of 1000, so: FishChips What is the probability that a randomly chosen person is eating fish? How many people are in the fish set? = 52. There are 100 people in total, so: S1: Venn Diagrams

Now for the last two diagrams: R G F What is the probability a person chosen at random has either red hair or wears glasses but not both? Add the numbers in the R circle that aren’t in the overlap with G, and the numbers in the G circle that aren’t in the overlap with R: = people in total, so: AX What is the probability that a student chosen at random has seen Avatar? 15 total in the A circle. 30 overall, so: S1: Venn Diagrams