1. Venn Diagrams © Christine Crisp “Teach A Level Maths” Statistics 1

2. Venn Diagrams "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages" Statistics 1 AQA EDEXCEL MEI/OCR OCR

3. Venn Diagrams Venn diagrams show the probabilities of more than one event and can be used instead of tree diagrams. They are quick and easy to use. Those of you taking the Edexcel or MEI specifications need to understand these diagrams. For the rest of you their use is optional but I am going to use them to illustrate some important laws of probability.

4. Venn Diagrams 20 e.g.1Amongst a group of 20 students, 7 are taking Maths and of these 3 are also taking Biology. 5 are taking neither. What is the probability that a student chosen at random is taking Biology? Solution: The “eggs” show Maths M B 5 43 The diagram shows the 20 students. and Biology 3 do both 5 do neither 7 do Maths ( but we have 3 already )

5. Venn Diagrams 20 e.g.1Amongst a group of 20 students, 7 are taking Maths and of these 3 are also taking Biology. 5 are taking neither. What is the probability that a student chosen at random is taking Biology? Solution: M B 5 43 8 The diagram shows the 20 students. The final number ( doing Biology but not Maths ) is given by So, P( student takes Biology ) =

6. Venn Diagrams 30 e.g.2 In a class of 30 students, 3 out of the 16 girls and 6 out of the 14 boys, are left-handed. Draw a Venn diagram and find the probability that a student chosen at random is a boy or left-handed. Solution: The diagram needs to show the numbers for Left-handedness and Boys. L B 6 8 3 13 There are 6 Left- handed Boys... so there are 8 boys who are not. There are 3 left- handed girls... and 13 who are right-handed.

7. Venn Diagrams 30 e.g.2 In a class of 30 students, 3 out of the 16 girls and 6 out of the 14 boys, are left-handed. Draw a Venn diagram and find the probability that a student chosen at random is a boy or left-handed. Solution: L B 6 8 3 13 P( boy or a left-hander ) = The diagram needs to show the numbers for Left-handedness and Boys.

8. Venn Diagrams Exercise 1. Customers at a restaurant are offered a choice of chips or jacket potato to go with either lasagna or pizza. Out of a group of 16, 11 have the lasagna. 7 choose chips to accompany their meal. 5 of those who choose chips have the lasagna. What is the probability that one chosen at random has neither chips nor lasagna? C L Solution: L: lasagna C: chips

9. Venn Diagrams Exercise 1. Customers at a restaurant are offered a choice of chips or jacket potato to go with either lasagna or pizza. Out of a group of 16, 11 have the lasagna. 7 choose chips to accompany their meal. 5 of those who choose chips have the lasagna. What is the probability that one chosen at random has neither chips nor lasagna? 16 C L Solution: L: lasagna C: chips

10. Venn Diagrams Exercise 1. Customers at a restaurant are offered a choice of chips or jacket potato to go with either lasagna or pizza. Out of a group of 16, 11 have the lasagna. 7 choose chips to accompany their meal. 5 of those who choose chips have the lasagna. What is the probability that one chosen at random has neither chips nor lasagna? 16C L 5 Solution: L: lasagna C: chips

11. Venn Diagrams Exercise 1. Customers at a restaurant are offered a choice of chips or jacket potato to go with either lasagna or pizza. Out of a group of 16, 11 have the lasagna. 7 choose chips to accompany their meal. 5 of those who choose chips have the lasagna. What is the probability that one chosen at random has neither chips nor lasagna? 16C L 5 6 Solution: L: lasagna C: chips

12. Venn Diagrams 1. Customers at a restaurant are offered a choice of chips or jacket potato to go with either lasagna or pizza. Out of a group of 16, 11 have the lasagna. 7 choose chips to accompany their meal. 5 of those who choose chips have the lasagna. What is the probability that one chosen at random has neither chips nor lasagna? 16C L 5 26 Solution: L: lasagna C: chips Exercise

13. Venn Diagrams 1. Customers at a restaurant are offered a choice of chips or jacket potato to go with either lasagna or pizza. Out of a group of 16, 11 have the lasagna. 7 choose chips to accompany their meal. 5 of those who choose chips have the lasagna. What is the probability that one chosen at random has neither chips nor lasagna? 16C L 5 26 3 Solution: L: lasagna C: chips Exercise

14. Venn Diagrams 1. Customers at a restaurant are offered a choice of chips or jacket potato to go with either lasagna or pizza. Out of a group of 16, 11 have the lasagna. 7 choose chips to accompany their meal. 5 of those who choose chips have the lasagna. What is the probability that one chosen at random has neither chips nor lasagna? 16C L 5 26 3 Solution: L: lasagna C: chips P(no chips, no lasagna) Exercise

15. Venn Diagrams We’ll use the example about left-handed students to illustrate a law of probability. 30 L B 6 83 13 I’ll change this to A as we usually use A when we state the law.

16. Venn Diagrams We’ll use the example about left-handed students to illustrate a law of probability. 30 A B 6 83 13

17. Venn Diagrams We’ll use the example about left-handed students to illustrate a law of probability. 30 A B 6 83 13 The number in A or B is anything shaded. N.B. A or B in probability includes both.

18. Venn Diagrams We’ll use the example about left-handed students to illustrate a law of probability. 30 A B 6 83 13 We’ll use n for “ the number in...”, so n( A or B ) = 3 + 6 + 8 = 17 n( A or B ) = 17

19. Venn Diagrams We’ll use the example about left-handed students to illustrate a law of probability. 30 A B 6 83 13 n( A ) = 9 n (B ) = n( A or B ) = 17 n( A ) = 9

20. Venn Diagrams n( A or B ) = 17 We’ll use the example about left-handed students to illustrate a law of probability. 30 A B 6 83 13 n (B ) = 14 n( A ) = 9 n( B ) = 14 n( A ) = 9

21. Venn Diagrams 30 A B 6 83 13 n( A or B ) = 17 We’ll use the example about left-handed students to illustrate a law of probability. The part with both types of shading gives the number in A and B. n( B ) = 14 n( A ) = 9 So, n( A and B ) = 6 n( A and B) = 6

22. Venn Diagrams We’ll use the example about left-handed students to illustrate a law of probability. 30 A B 6 83 13 n( A or B ) = 17 n( B ) = 14 n( A ) = 9 n( A )  n( A and B) = 6 We now get n( A or B ) = 17 = n( A and B )n( B )  9  14  6 N.B. The part with both types of shading is in A and in B so it has been counted twice. We subtract one lot.

23. Venn Diagrams We’ll use the example about left-handed students to illustrate a law of probability. 30 A B 6 83 13 n( A or B ) = 17 n( B ) = 14 n( A ) = 9 n( A )  n( A and B) = 6 We now get n( A or B ) = n( A and B )n( B )  Dividing by 30 gives probabilities, so P( A or B ) = P( A ) + P( B ) – P( A and B )

24. Venn Diagrams A B If A and B don’t overlap, P( A and B ) = 0 ( the intersection is empty ) P( A or B ) = P( A ) + P( B ) – P( A and B ) So, P( A or B ) = P( A ) + P( B ) A and B are said to be mutually exclusive events. ( If A happens, B cannot or if B happens, A cannot. )

25. Venn Diagrams Notation We can write P (A or B) as P (A  B) P ( A or B ) = P(A ) + P(B ) – P ( A and B ) so, I remember the notation by thinking of this symbol as a cup which can hold anything in A or B. and P (A and B) as P (A  B) becomes P ( A  B ) = P ( A ) + P ( B ) – P ( A  B ) means “ the probability that event A does not occur ” Also, P ( A / )

26. Venn Diagrams  If A and B are 2 events SUMMARY  If P (A and B ) = 0, A and B are mutually exclusive and then, P (A or B) can be written as P (A B)  P (A and B) can be written as P (A B)  P( A or B ) = P( A ) + P( B ) – P( A and B ) P( A or B ) = P(A) + P( B )

27. Venn Diagrams Solution: P(A or B) = P (A) + P (B) – P (A and B) e.g.1 Events A and B are such that P (A) P (B) and Find P (A and B). P (A or B) P (A and B) 

28. Venn Diagrams (a) We want to find P (P or Y) Solution: P ( P or Y ) = P (P) + P (Y) – P (P and Y) e.g.2 In a group of 15 mixed plants, 6 are petunias, 8 are yellow and 3 are both. What is the probability if a plant is picked at random that it is (a) either a petunia or yellow (b) either a petunia or yellow but not both? Let P be event ”Petunia” and Y be event “Yellow”

29. Venn Diagrams (a) We want to find P (P or Y) Solution: Let P be event ”Petunia” and Y be event “Yellow” 6 P ( P or Y ) = P (P) + P (Y) – P (P and Y) e.g.2 In a group of 15 mixed plants, 6 are petunias, 8 are yellow and 3 are both. What is the probability if a plant is picked at random that it is (a) either a petunia or yellow (b) either a petunia or yellow but not both? Solution: Let P be event ”Petunia” and Y be event “Yellow”

30. Venn Diagrams (a) We want to find P (P or Y) Solution: Let P be event ”Petunia” and Y be event “Yellow” 68 P ( P or Y ) = P (P) + P (Y) – P (P and Y) e.g.2 In a group of 15 mixed plants, 6 are petunias, 8 are yellow and 3 are both. What is the probability if a plant is picked at random that it is (a) either a petunia or yellow (b) either a petunia or yellow but not both? Solution: Let P be event ”Petunia” and Y be event “Yellow”

31. Venn Diagrams (a) We want to find P (P or Y) Solution: Let P be event ”Petunia” and Y be event “Yellow” 683 P ( P or Y ) = P (P) + P (Y) – P (P and Y) e.g.2 In a group of 15 mixed plants, 6 are petunias, 8 are yellow and 3 are both. What is the probability if a plant is picked at random that it is (a) either a petunia or yellow (b) either a petunia or yellow but not both? Solution: Let P be event ”Petunia” and Y be event “Yellow”

32. Venn Diagrams e.g.2 In a group of 15 mixed plants, 6 are petunias, 8 are yellow and 3 are both. What is the probability if a plant is picked at random that it is (a) either a petunia or yellow (b) either a petunia or yellow but not both? Solution: Let P be event ”Petunia” and Y be event “Yellow” 683 P ( P or Y ) = P (P) + P (Y) – P (P and Y) (a) We want to find P (P or Y) A Venn diagram can also be used to answer the question. (b) We must subtract another P (P and Y). ANS:

33. Venn Diagrams Solution: Let P be event ”Petunia” and Y be event “Yellow” Y P (a) We want to find P (P or Y) e.g.2 In a group of 15 mixed plants, 6 are petunias, 8 are yellow and 3 are both. What is the probability if a plant is picked at random that it is (a) either a petunia or yellow (b) either a petunia or yellow but not both?

34. Venn Diagrams 15 Solution: Let P be event ”Petunia” and Y be event “Yellow” Y P (a) We want to find P (P or Y) e.g.2 In a group of 15 mixed plants, 6 are petunias, 8 are yellow and 3 are both. What is the probability if a plant is picked at random that it is (a) either a petunia or yellow (b) either a petunia or yellow but not both?

35. Venn Diagrams 15 Solution: Let P be event ”Petunia” and Y be event “Yellow” Y P 3 (a) We want to find P (P or Y) e.g.2 In a group of 15 mixed plants, 6 are petunias, 8 are yellow and 3 are both. What is the probability if a plant is picked at random that it is (a) either a petunia or yellow (b) either a petunia or yellow but not both?

36. Venn Diagrams 15 Solution: Let P be event ”Petunia” and Y be event “Yellow” Y P 3 3 (a) We want to find P (P or Y) e.g.2 In a group of 15 mixed plants, 6 are petunias, 8 are yellow and 3 are both. What is the probability if a plant is picked at random that it is (a) either a petunia or yellow (b) either a petunia or yellow but not both?

37. Venn Diagrams 15 Solution: Let P be event ”Petunia” and Y be event “Yellow” Y P 35 3 (a) We want to find P (P or Y) e.g.2 In a group of 15 mixed plants, 6 are petunias, 8 are yellow and 3 are both. What is the probability if a plant is picked at random that it is (a) either a petunia or yellow (b) either a petunia or yellow but not both?

38. Venn Diagrams 15 Solution: Let P be event ”Petunia” and Y be event “Yellow” Y P 35 3 4 P (P or Y) = (a) We want to find P (P or Y) e.g.2 In a group of 15 mixed plants, 6 are petunias, 8 are yellow and 3 are both. What is the probability if a plant is picked at random that it is (a) either a petunia or yellow (b) either a petunia or yellow but not both?

39. Venn Diagrams 15 Solution: Let P be event ”Petunia” and Y be event “Yellow” Y P 35 3 4 P (P or Y) = (a) We want to find P (P or Y) e.g.2 In a group of 15 mixed plants, 6 are petunias, 8 are yellow and 3 are both. What is the probability if a plant is picked at random that it is (a) either a petunia or yellow (b) either a petunia or yellow but not both? (b)

40. Venn Diagrams 15 Solution: Let P be event ”Petunia” and Y be event “Yellow” Y P 35 3 4 P (P or Y) = (a) We want to find P (P or Y) e.g.2 In a group of 15 mixed plants, 6 are petunias, 8 are yellow and 3 are both. What is the probability if a plant is picked at random that it is (a) either a petunia or yellow (b) either a petunia or yellow but not both? (b)

41. Venn Diagrams A B p e.g.3 A, B are 2 events such that Draw a Venn diagram and use it to help you to find and Solution: Let

42. Venn Diagrams e.g.3 A, B are 2 events such that A B p Solution: Let and e.g.3 A, B are 2 events such that Draw a Venn diagram and use it to help you to find and

43. Venn Diagrams A B p Solution: Let and e.g.3 A, B are 2 events such that Draw a Venn diagram and use it to help you to find and

44. Venn Diagrams A B p Solution: Let and Finally, ( probability of not in A and not in B ) The total probability is 1, so e.g.3 A, B are 2 events such that Draw a Venn diagram and use it to help you to find and

45. Venn Diagrams 1. Events A and B are such that P (B) P (A or B) and P (A and B)Find P (A). 2. In a group of 20 students, 8 play music, 11 belong to a sports team and 6 do both. What is the probability that a student picked at random from the group plays music or belongs to a sports team? Exercise

46. Venn Diagrams P (A or B) = P (A) + P (B) – P (A and B) P (A) + P (A) = Solutions: Either: Or: A B 1. Events A and B are such that P (B) , P (A or B)  Find P (A). and P (A and B) .

47. Venn Diagrams and P (A and B) . Find P (A). P (A or B) = P (A) + P (B) – P (A and B) P (A) + P (A) = Solutions: Or: A B 1. Events A and B are such that P (B) , P (A or B)  Either:

48. Venn Diagrams 1. Events A and B are such that P (B) , P (A or B)  P (A or B) = P (A) + P (B) – P (A and B) P (A) + P (A) = Solutions: Or: A B Find P (A). and P (A and B) . Either:

49. Venn Diagrams P (A or B) = P (A) + P (B) – P (A and B) P (A) + P (A) = Solutions: Or: A B 1. Events A and B are such that P (B) , P (A or B)  Find P (A). and P (A and B) . Either:

50. Venn Diagrams P (A or B) = P (A) + P (B) – P (A and B) P (A) + P (A) = Solutions: Or: A B P (A) = 1. Events A and B are such that P (B) , P (A or B)  Find P (A). and P (A and B) . Either:

51. Venn Diagrams Solution: 2. In a group of 20 students, 8 play music, 11 belong to a sports team and 6 do both. What is the probability that a student picked at random from the group plays music or belongs to a sports team? Let M be event “plays music” and S “is in sports team” P (M or S) = P (M) + P (S) – P (M and S) Either: Exercise

52. Venn Diagrams 20 S M Solution: 6 52 7 Let M be event “plays music” and S “is in sports team” P (M or S) Exercise Or: 2. In a group of 20 students, 8 play music, 11 belong to a sports team and 6 do both. What is the probability that a student picked at random from the group plays music or belongs to a sports team?

54. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

55. Venn Diagrams Venn diagrams show the probabilities of more than one event and can be used instead of tree diagrams. They are quick and easy to use. Those of you taking the Edexcel or MEI specifications need to understand these diagrams. For the rest of you their use is optional but I am going to use them to illustrate some important laws of probability.

56. Venn Diagrams 20 e.g.1Amongst a group of 20 students, 7 are taking Maths and of these 3 are also taking Biology. 5 are taking neither. What is the probability that a student chosen at random is taking Biology? Solution: The “eggs” show Maths M B 5 43 The diagram shows the 20 students. and Biology So, P( student takes Biology ) = The number doing Biology but not Maths is given by 8

57. Venn Diagrams 30 e.g.2 In a class of 30 students, 3 out of the 16 girls and 6 out of the 14 boys, are left-handed. Draw a Venn diagram and find the probability that a student chosen at random is a boy or left-handed. Solution: L B 6 8 3 13 P( boy or a left-hander ) = The diagram needs to show the numbers for Left-handedness and Boys. There are 6 left- handed boys so 8 boys are not. There are 3 left- handed girls and 13 right-handed

58. Venn Diagrams We’ll use the example about left-handed students to illustrate a law of probability. 30 A B 3 13 n( A or B ) = 17 n( B ) = 14 n( A ) = 9 n( A )  n( A and B) = 6 We now get n( A or B ) = n( A and B )n( B )  Dividing by 30 gives probabilities, so P( A or B ) = P( A ) + P( B ) – P( A and B ) 8 6

59. Venn Diagrams  If A and B are 2 events SUMMARY  If P (A and B ) = 0, A and B are mutually exclusive and then, P (A or B) can be written as P (A B)  P (A and B) can be written as P (A B)  P( A or B ) = P( A ) + P( B ) – P( A and B ) P( A or B ) = P( A ) + P( B )