1. © Christine Crisp The following slides show one of the 40 presentations that cover the A/AS Level Mathematics option module S1. Teach A Level Maths Vol 3: Statistics 1 Demo Disc

2. Conditional Probability © Christine Crisp Teach A Level Maths Statistics 1

3. Conditional Probability "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

4. Conditional Probability We talk about conditional probability when the probability of one event depends on whether or not another event has occurred. Conditional probability problems can be solved by considering the individual possibilities or by using a table, a Venn diagram, a tree diagram or a formula. Harder problems are most easily solved by using a formula together with a tree diagram. e.g. There are 2 red and 3 blue counters in a bag and, without looking, we take out one counter and do not replace it. The probability of a 2 nd counter taken from the bag being red depends on whether the 1 st was red or blue.

5. Conditional Probability Notation P (A) means the probability that event A occurs P (A / ) means the probability that event A does not occur the probability that event A occurs given that B has occurred. This is conditional probability. P (A B) means

6. Conditional Probability e.g. 1. The following table gives data on the type of car, grouped by petrol consumption, owned by 100 people. One person is selected at random. 42123 Female 73312 Male HighMediumLow 100 Total L is the event the person owns a low rated car

7. Conditional Probability Female Low Male e.g. 1. The following table gives data on the type of car, grouped by petrol consumption, owned by 100 people. 42123 73312 HighMedium One person is selected at random. L is the event the person owns a low rated car F is the event a female is chosen. 100 Total

8. Conditional Probability Female Low Male e.g. 1. The following table gives data on the type of car, grouped by petrol consumption, owned by 100 people. 42123 73312 HighMedium One person is selected at random. L is the event the person owns a low rated car F is the event a female is chosen. 100 Total

9. Conditional Probability Female Low Male e.g. 1. The following table gives data on the type of car, grouped by petrol consumption, owned by 100 people. 42123 73312 HighMedium One person is selected at random. L is the event the person owns a low rated car F is the event a female is chosen. There is no need for a Venn diagram or a formula to solve this type of problem. We just need to be careful which row or column we look at. Find (i) P (L) (ii) P (F and L)(iii) P (F L) 100 Total

10. Conditional Probability 23 12 100 (i) P (L) = Solution: Find (i) P (L) (ii) P (F and L)(iii) P (F L) 35 421Female 733Male TotalHighMediumLow ( I usually leave the answers as fractions as they wont always be exact decimals. Its good practice to cancel. ) 23 12 Low

11. Conditional Probability (i) P (L) = Solution: Find (i) P (L) (ii) P (F and L)(iii) P (F L) 100 42123Female 73312Male HighMediumLow (ii) P (F and L) = The probability of selecting a female with a low rated car. 23 100 Total

12. Conditional Probability (i) P (L) = Solution: Find (i) P (L) (ii) P (F and L)(iii) P (F L) 10035 42123Female 73312Male HighMediumLow (ii) P (F and L) = Total (iii) P (F L) The probability of selecting a female given the car is low rated. We must be careful with the denominators in (ii) and (iii). Here we are given the car is low rated. We want the total of that column. 23 12

13. Conditional Probability (i) P (L) = Solution: Find (i) P (L) (ii) P (F and L)(iii) P (F L) 100 42123Female 73312Male HighMediumLow (ii) P (F and L) = Total (iii) P (F L) Notice that P (L) P (F L) So, P (F and L) = P(F L) P (L) = P (F and L)

14. Conditional Probability However, I havent proved the formula, just shown that it works for one particular problem. This result can be used to help solve harder conditional probability problems. Well just illustrate it again on a simple problem using a Venn diagram. P (F and L) = P(F L) P (L)

15. Conditional Probability e.g. 2. I have 2 packets of seeds. One contains 20 seeds and although they look the same, 8 will give red flowers and 12 blue. The 2nd packet has 25 seeds of which 15 will be red and 10 blue. Draw a Venn diagram and use it to illustrate the conditional probability formula. Solution:LetR be the event Red flower and F be the event First packet R F Red in the 1 st packet

16. Conditional Probability e.g. 2. I have 2 packets of seeds. One contains 20 seeds and although they look the same, 8 will give red flowers and 12 blue. The 2nd packet has 25 seeds of which 15 will be red and 10 blue. Draw a Venn diagram and use it to illustrate the conditional probability formula. Solution: R F 8 Red in the 1 st packet LetR be the event Red flower and F be the event First packet

17. Conditional Probability e.g. 2. I have 2 packets of seeds. One contains 20 seeds and although they look the same, 8 will give red flowers and 12 blue. The 2nd packet has 25 seeds of which 15 will be red and 10 blue. Draw a Venn diagram and use it to illustrate the conditional probability formula. Solution: R F Blue in the 1 st packet 8 LetR be the event Red flower and F be the event First packet

18. Conditional Probability e.g. 2. I have 2 packets of seeds. One contains 20 seeds and although they look the same, 8 will give red flowers and 12 blue. The 2nd packet has 25 seeds of which 15 will be red and 10 blue. Draw a Venn diagram and use it to illustrate the conditional probability formula. Solution: R F Blue in the 1 st packet 812 LetR be the event Red flower and F be the event First packet

19. Conditional Probability e.g. 2. I have 2 packets of seeds. One contains 20 seeds and although they look the same, 8 will give red flowers and 12 blue. The 2nd packet has 25 seeds of which 15 will be red and 10 blue. Draw a Venn diagram and use it to illustrate the conditional probability formula. Solution: R F Red in the 2 nd packet 812 LetR be the event Red flower and F be the event First packet

20. Conditional Probability e.g. 2. I have 2 packets of seeds. One contains 20 seeds and although they look the same, 8 will give red flowers and 12 blue. The 2nd packet has 25 seeds of which 15 will be red and 10 blue. Draw a Venn diagram and use it to illustrate the conditional probability formula. Solution: R F Red in the 2 nd packet 15 812 LetR be the event Red flower and F be the event First packet

21. Conditional Probability e.g. 2. I have 2 packets of seeds. One contains 20 seeds and although they look the same, 8 will give red flowers and 12 blue. The 2nd packet has 25 seeds of which 15 will be red and 10 blue. Draw a Venn diagram and use it to illustrate the conditional probability formula. Solution: R F 15 812 Blue in the 2 nd packet LetR be the event Red flower and F be the event First packet

22. Conditional Probability e.g. 2. I have 2 packets of seeds. One contains 20 seeds and although they look the same, 8 will give red flowers and 12 blue. The 2nd packet has 25 seeds of which 15 will be red and 10 blue. Draw a Venn diagram and use it to illustrate the conditional probability formula. Solution: R F 15 10 812 Blue in the 2 nd packet LetR be the event Red flower and F be the event First packet

23. Conditional Probability e.g. 2. I have 2 packets of seeds. One contains 20 seeds and although they look the same, 8 will give red flowers and 12 blue. The 2nd packet has 25 seeds of which 15 will be red and 10 blue. Draw a Venn diagram and use it to illustrate the conditional probability formula. Solution: R F 15 10 812 Total: 20 + 25 LetR be the event Red flower and F be the event First packet

24. Conditional Probability 45 e.g. 2. I have 2 packets of seeds. One contains 20 seeds and although they look the same, 8 will give red flowers and 12 blue. The 2nd packet has 25 seeds of which 15 will be red and 10 blue. Draw a Venn diagram and use it to illustrate the conditional probability formula. Solution: R F 15 10 812 Total: 20 + 25 LetR be the event Red flower and F be the event First packet

25. Conditional Probability 45 e.g. 2. I have 2 packets of seeds. One contains 20 seeds and although they look the same, 8 will give red flowers and 12 blue. The 2nd packet has 25 seeds of which 15 will be red and 10 blue. Draw a Venn diagram and use it to illustrate the conditional probability formula. Solution: R F 15 10 812 LetR be the event Red flower and F be the event First packet

26. Conditional Probability 45 R F e.g. 2. I have 2 packets of seeds. One contains 20 seeds and although they look the same, 8 will give red flowers and 12 blue. The 2nd packet has 25 seeds of which 15 will be red and 10 blue. Draw a Venn diagram and use it to illustrate the conditional probability formula. Solution: 15 10 812 P (R and F) = LetR be the event Red flower and F be the event First packet

27. Conditional Probability 45 R F e.g. 2. I have 2 packets of seeds. One contains 20 seeds and although they look the same, 8 will give red flowers and 12 blue. The 2nd packet has 25 seeds of which 15 will be red and 10 blue. Draw a Venn diagram and use it to illustrate the conditional probability formula. Solution: 15 10 812 P (R and F) = LetR be the event Red flower and F be the event First packet

28. Conditional Probability 45 R F e.g. 2. I have 2 packets of seeds. One contains 20 seeds and although they look the same, 8 will give red flowers and 12 blue. The 2nd packet has 25 seeds of which 15 will be red and 10 blue. Draw a Venn diagram and use it to illustrate the conditional probability formula. Solution: 15 10 812 P (R and F) = LetR be the event Red flower and F be the event First packet

29. Conditional Probability 45 R F e.g. 2. I have 2 packets of seeds. One contains 20 seeds and although they look the same, 8 will give red flowers and 12 blue. The 2nd packet has 25 seeds of which 15 will be red and 10 blue. Draw a Venn diagram and use it to illustrate the conditional probability formula. Solution: 15 10 12 P (R and F) = P (R F) = 8 LetR be the event Red flower and F be the event First packet

30. Conditional Probability 45 R F e.g. 2. I have 2 packets of seeds. One contains 20 seeds and although they look the same, 8 will give red flowers and 12 blue. The 2nd packet has 25 seeds of which 15 will be red and 10 blue. Draw a Venn diagram and use it to illustrate the conditional probability formula. Solution: 15 10 12 P (R and F) = P (F) =P (R F) = 8 LetR be the event Red flower and F be the event First packet

31. Conditional Probability 45 R F e.g. 2. I have 2 packets of seeds. One contains 20 seeds and although they look the same, 8 will give red flowers and 12 blue. The 2nd packet has 25 seeds of which 15 will be red and 10 blue. Draw a Venn diagram and use it to illustrate the conditional probability formula. Solution: 15 10 12 P (R and F) = P (F) =P (R F) = 8 LetR be the event Red flower and F be the event First packet

32. Conditional Probability 45 R F e.g. 2. I have 2 packets of seeds. One contains 20 seeds and although they look the same, 8 will give red flowers and 12 blue. The 2nd packet has 25 seeds of which 15 will be red and 10 blue. Draw a Venn diagram and use it to illustrate the conditional probability formula. Solution: 15 10 12 P (R and F) = P (F) =P (R F) = 8 LetR be the event Red flower and F be the event First packet

33. Conditional Probability 45 R F e.g. 2. I have 2 packets of seeds. One contains 20 seeds and although they look the same, 8 will give red flowers and 12 blue. The 2nd packet has 25 seeds of which 15 will be red and 10 blue. Draw a Venn diagram and use it to illustrate the conditional probability formula. Solution: 15 10 12 P (R and F) = P (F) =P (R F) = 8 P (R and F) = P(R F) P (F) So, P (R F) P (F) = LetR be the event Red flower and F be the event First packet

34. Conditional Probability SUMMARY The probability that both event A and event B occur is given by P (A and B) can also be written as P (A B) We often use this in the form Reminder: In words, this is the probability of event A given that B has occurred, equals the probability of both A and B occurring divided by the probability of B. P ( A B ) P ( A and B ) P(B)P(B) P ( A and B ) = P (A B) P (B)

35. Conditional Probability e.g. 3. In November, the probability of a man getting to work on time if there is fog on the M6 is. If the visibility is good, the probability is. The probability of fog at the time he travels is. (a)Calculate the probability of him arriving on time. There are lots of clues in the question to tell us we are dealing with conditional probability. (b)Calculate the probability that there was fog given that he arrives on time.

36. Conditional Probability e.g. 3. In November, the probability of a man getting to work on time if there is fog on the M6 is. If the visibility is good, the probability is. The probability of fog at the time he travels is. (a)Calculate the probability of him arriving on time. (b)Calculate the probability that there was fog given that he arrives on time. There are lots of clues in the question to tell us we are dealing with conditional probability. Solution:Let T be the event getting to work on time Let F be the event fog on the M6 Can you write down the notation for the probabilities that we want to find in (a) and (b)?

37. Conditional Probability the probability of a man getting to work on time if there is fog is If the visibility is good, the probability is. The probability of fog at the time he travels is. Can you also write down the notation for the three probabilities given in the question? This is a much harder problem so we draw a tree diagram. (a)Calculate the probability of him arriving on time. P (F T) P (T) P (T F) P (T F / ) P (F) (b)Calculate the probability that there was fog given that he arrives on time. Not foggy

38. Conditional Probability Not on time Fog No Fog On time Not on time P (T F) P (T F / ) P (F) F F/F/ T T/T/ T T/T/

39. Conditional Probability P (T F) P (T F / ) P (F) F F/F/ T T/T/ T T/T/ Because we only reach the 2 nd set of branches after the 1 st set has occurred, the 2 nd set must represent conditional probabilities.

40. Conditional Probability (a)Calculate the probability of him arriving on time. F F/F/ T T/T/ T T/T/

41. Conditional Probability F F/F/ T T/T/ T T/T/ (a)Calculate the probability of him arriving on time. ( foggy and he arrives on time )

42. Conditional Probability F F/F/ T T/T/ T T/T/ (a)Calculate the probability of him arriving on time. ( not foggy and he arrives on time )

43. Conditional Probability Fog on M 6 Getting to work F T From part (a), (b)Calculate the probability that there was fog given that he arrives on time. We need

44. Conditional Probability Exercise 1. A sample of 100 adults were asked how they travelled to work. The results are shown in the table. WalkBusBikeCar Men1012626 Women718417 Total100 Find (i) (ii) (iii) (iv) P (M) P (M / and C) One person is picked at random. M is the event the person is a man C is the event the person travels by car P (M C) P (C M / )

45. Conditional Probability 100 54 Total 174187Women 2661210Men CarBikeBusWalk Find (i) (ii) (iii) (iv) P (M) P (M / and C) P (M C) P (C M / ) Solution: (i) P (M) (i) (ii) P (M C)

46. Conditional Probability 10043Total 174187Women 542661210Men CarBikeBusWalk Find (i) (ii) (iii) (iv) P (M) P (M / and C) P (M C) P (C M / ) Solution: (i) P (M) (ii) P (M C) (iii) P (M / and C)

47. Conditional Probability 100 43Total 174187Women 542661210Men CarBikeBusWalk Find (i) (ii) (iii) (iv) P (M) P (M / and C) P (M C) P (C M / ) Solution: (i) P (M) (ii) P (M C) (iii) P (M / and C) P (C M / ) (iv)

48. Conditional Probability 10043Total 46174187Women 542661210Men CarBikeBusWalk Find (i) (ii) (iii) (iv) P (M) P (M / and C) P (M C) P (C M / ) Solution: (i) P (M) (ii) P (M C) (iii) P (M / and C) P (C M / ) (iv)

49. Conditional Probability Exercise 2. The probability of a maximum temperature of 28 or more on the 1 st day of Wimbledon ( tennis competition! ) has been estimated as. The probability of a particular British player winning on the 1 st day if it is below 28 is estimated to be but otherwise only. Draw a tree diagram and use it to help solve the following: (i)the probability of the player winning, (ii)the probability that, if the player has won, it was at least 28. Solution: Let T be the event temperature 28 or more Let W be the event player wins Then,

50. Conditional Probability High temp W Wins Loses Lower temp Sum = 1 Let T be the event temperature 28 or more Let W be the event player wins Then, T T/T/ Wins W W/W/ W/W/

51. Conditional Probability (i) W T T/T/ W W/W/ W/W/

52. Conditional Probability (i) W T T/T/ W W/W/ W/W/

53. Conditional Probability W T T/T/ W W/W/ W/W/ (ii)

54. Conditional Probability (ii) W T T/T/ W W/W/ W/W/

55. Conditional Probability (ii) W T T/T/ W W/W/ W/W/

56. Conditional Probability We can deduce an important result from the conditional law of probability: ( The probability of A does not depend on B. ) or P ( A and B ) P ( A ) P ( B ) If B has no effect on A, then, P (A B) = P (A) and we say the events are independent. becomes P ( A ) P ( A and B ) P(B)P(B) So, P ( A B ) P ( A and B ) P(B)P(B)

57. Conditional Probability SUMMARY For 2 independent events, P ( A and B ) P ( A ) P ( B ) P ( A B ) P ( A ) So,

59. 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.

60. Conditional Probability SUMMARY means the probability that event A occurs given that B has occurred means the probability that event A occurs means the probability that event A does not occur Reminder: P (A and B) can also be written as In words, this is the probability of event A given that B has occurred, equals the probability of both A and B occurring divided by the probability of B. Rearranging:

61. Conditional Probability Female Low Male e.g. 1. The following table gives data on the type of car, grouped by petrol consumption, owned by 100 people. 42123 73312 HighMedium One person is selected at random. L is the event the person owns a low rated car F is the event a female is chosen. There is no need for a Venn diagram or a formula to solve this type of problem. We just need to be careful which row or column we look at. Find (i) P (L) (ii) P (F and L)(iii) P (F L) 100 Total

62. Conditional Probability (i) P (L) = Solution: Find (i) P (L) (ii) P (F and L)(iii) P (F L) 100 42123Female 73312Male HighMediumLow (ii) P (F and L) = Total (iii) P (F L) Notice that P (L) P (F L) = P (F and L) So, P (F and L) = P(F L) P (L)

63. Conditional Probability 45 R F e.g. 2. I have 2 packets of seeds. One contains 20 seeds and although they look the same, 8 will give red flowers and 12 blue. The 2nd packet has 25 seeds of which 15 will be red and 10 blue. Draw a Venn diagram and use it to illustrate the conditional probability formula. Solution:LetR be the event Red flower and F be the event First packet 15 10 12 P (R and F) = P (F) =P (R F) = 8 P (R and F) = P(R F) P (F) So, P (R F) P (F) =

64. Conditional Probability e.g. 3. In November, the probability of a man getting to work on time if there is fog on the M6 is. If the visibility is good, the probability is. The probability of fog at the time he travels is. (a)Calculate the probability of him arriving on time. (b)Calculate the probability that there was fog given that he arrives on time. Solution: Let T be the event getting to work on time Let F be the event fog on the M6

65. Conditional Probability F F/F/ T T/T/ T T/T/ (a)Calculate the probability of him arriving on time.

66. Conditional Probability Fog on M 6 Getting to work F T From part (a), (b)Calculate the probability that there was fog given that he arrives on time. We need

67. Conditional Probability So, for 2 independent events, ( The probability of A does not depend on B. ) If B has no effect on A, then, P (A B) = P (A) and we say the events are independent.