Win if you get a RED!

1. The British actor Anthony Hopkins was delighted to hear that he had landed a leading role in a film based on the book The Girl From Petrovka by George Feifer. A few days after signing the contract, Hopkins travelled to London to buy a copy of the book. He tried several bookshops, but there wasn't one to be had. Waiting at Leicester Square underground for his train home, he noticed a book apparently discarded on a bench. Incredibly, it was The Girl From Petrovka. That in itself would have been coincidence enough but in fact it was merely the beginning of an extraordinary chain of events. Two years later, in the middle of filming in Vienna, Hopkins was visited by George Feifer, the author. Feifer mentioned that he did not have a copy of his own book. He had lent the last one - containing his own annotations - to a friend who had lost it somewhere in London. With mounting astonishment, Hopkins handed Feifer the book he had found. 'Is this the one?' he asked, 'with the notes scribbled in the margins?' It was the same book.

2. Win if you get a RED! Would you play this game? What is the probability of a red? B P(red) = 2/8 = 1/4 G G What is the probability of not getting a red? R B P(not a red) = 1 – P(red) = 1 – ¼ = ¾ B R What is the probability of getting a red or a green? B P(r or a g) = P(red) + P(green) = 2/8 + 2/8 = 4/8 = 1/2

3. Lesson Objective Understand that probability is a measure of how likely something is to happen and be able to calculate probabilities for a single event Know that P(something does not happen) = 1 – P(It does happen)

4. When a situation has several equally likely outcomes it is possible to calculate the probability of an outcome occurring by using the formula: Probability (event) = No. of ways an event can happen Total number of all possible outcomes This will give a value between 0 and 1, where 0 is impossible and 1 is certain. Probability can either be expressed asa fraction, decimal or a percentage. We will use FRACTIONS, occasionally decimals. Impossible Unlikely Even Chance Likely Certain 0 ¾ 1 ¼ ½

6. 1a) ½ b) ½ c) ¼ d) 3/8

7. What if we have a more complicated situation? I Spin the spinner twice and I only win if I get the exactly the same colour on both spins. B G G R B B R B

8. Spin 2 B G G R B Spin 1 B R B

9. Spin 2 B G G R B Spin 1 B R B

10. To calculate the probability of an event we need to consider all the equally likely outcomes. The list of equally likely outcomes is called the POSSIBILITY SPACE Then we can use the formula: Probability (event) = No. of ways an event can happen Total number of all possible outcomes

11. Lotto Choose 4 numbers. The lotto numbers are going to be created by rolling 2 dice and adding the resulting total of each of the three numbers together; so choose your numbers wisely. The first to get a line wins the game.

12. Assuming that the dice really are random and fair, which numbers were the best to choose and why?

13. 2nd die + 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 1st die 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 Assuming that the dice really are random and fair, which numbers were the best to choose and why? There are 36 possible combinations. P(2) = 1/36 P(3) = 2/36 = 1/18 P(4) = 3/36 = 1/12 etc You can clearly see that the best numbers to chose are 5,6,7 and 8 or 6,7,8 and 9.

14. Lesson Objective Be able to use a Possibility Space Diagram to calculate the probability of combined events

15. In my family there are two adults: Me, my partner Carol and my two children Poppy and Lucy. Fun Maths We sit in a line on a park bench to have a picnic. Assuming that we sit down randomly, what is the probability that the two children end up sitting next to each other?

16. In my family there are two adults: Me, my partner Carol and my two children Poppy and Lucy. Fun Maths We sit in a line on a park bench to have a picnic. Assuming that we sit down randomly, what is the probability that the two children end up sitting next to each other? Suppose Granny Annie joins us and we have 5 people, what is the probability that the children sit next to each other?

17. We need to draw up a list of the possible outcomes (the possibility space) MWLP MWPL MLWP MLPW MPLW MPWL MWLP MWPL MLWP MLPW MPLW MPWL WMLP WMPL WPML WPLM WLPM WLMP WMLP WMPL WPML WPLM WLPM WLMP LWPM LWMP LPMW LPWM LMPW LMWP LWPM LWMP LPMW LPWM LMPW LMWP PWLM PWML PMLW PMWL PLWM PLMW PWLM PWML PMLW PMWL PLWM PLMW There are 24 ways we can sit on the bench, so 24 possible outcomes in the possibility space. In 12 of them the children sit together so 12/24 = 1/2

18. When a situation has several equally likely outcomes it is possible to calculate the probability of an outcome occurring by using the formula: Probability (event) = No. of ways an event can happen Total number of all possible outcomes Eg When I flip 3 fair coins what is the probability that I get 3 Heads?

19. Second die + 1 2 3 4 1 2 3 4 5 2 3 4 5 6 First die 3 4 5 6 7 4 5 6 7 8 Two four-sided dice are thrown and the numbers added together. Construct a sample space diagram to show all the outcomes. • What is the probability of getting: • a total more than 4? • a total less than 8? • a prime number total? • a total that is at least 3? • a total of 4 or 5? • the same number on both dice? • a lower number on the first dice?

22. Consider the following situation: I roll two six sided die and look at the difference in the scores. What is the probability that the difference in the scores is a square number?

23. Lesson Objective Understand when two events are mutually exclusive Learn that when two events, A and B are mutually exclusive we can use the formula P(A OR B) = P(A) + P(B)

24. Two events are mutually exclusive if they do not overlap Eg A = I pick a male B = I pick a female Eg A = On a fair coin I flip a Head B = On a fair coin I flip a Tail

25. Stand up if you think these ARE MUTUALLY EXCLUSIVE On a fair die A: Roll an even number B: Roll an odd number

26. Stand up if you think these ARE MUTUALLY EXCLUSIVE On a fair die A: A randomly chosen word begins with the letter ‘a’ B: A randomly chosen word begins with the letter ‘b’

27. Stand up if you think these ARE MUTUALLY EXCLUSIVE On a fair die A: You roll a prime number B: You roll an even number

28. Stand up if you think these ARE MUTUALLY EXCLUSIVE When you look at a light A: The light is on B: The light is off

29. Stand up if you think these ARE MUTUALLY EXCLUSIVE When you look outside A: It is sunny B: It is windy

30. Stand up if you think these ARE MUTUALLY EXCLUSIVE A: You are male B: You are in the top set

31. Write down your own example of a mutually exclusive pair of events

32. Second die + 1 2 3 4 1 2 2 3 3 3 4 4 4 5 5 5 2 3 3 3 4 4 4 5 5 5 6 6 6 First die 3 4 4 4 5 5 5 6 6 6 7 7 P(3 or 4) = P(3) + P(4) = 4 5 5 5 6 6 6 7 7 8 1 3 5 2 2 3 16 16 16 16 16 16 So you can find this probability by simply adding the two separate probabilities. + = + = Similarly, P(2 or 7) = P(2) + P(7) = Use the table to find the probability of getting a score of 3 or 4. The probability of getting a score of 3 or 4 can be written as P(3 or 4). Notice that:

33. + 1 2 3 4 5 6 1 2 3 3 3 4 4 4 5 5 6 6 7 2 3 3 3 4 4 4 5 5 6 6 7 8 P(3 or 4) = P(3) + P(4) 3 4 4 4 5 5 6 6 7 8 9 = 4 5 5 6 6 7 8 9 10 5 3 2 4 5 9 36 36 36 36 36 36 5 6 6 7 8 9 10 11 += + = 6 7 8 9 10 11 12 = 1 4 Two six-sided dice are thrown. Work out P(3 or 4) by adding fractions. Work out P(5 or 6) by adding fractions. P(5 or 6) = P(5) + P(6) =

34. + 1 2 3 4 5 6 1 2 2 2 3 3 4 4 5 5 6 6 7 7 2 3 3 4 4 5 5 6 6 7 7 8 8 3 4 4 5 5 6 6 7 7 8 8 9 4 5 5 6 6 7 7 8 8 9 10 10 5 6 6 7 7 8 8 9 10 10 11 11 6 7 7 8 8 9 10 10 11 11 12 12 P(a prime number OR an even number) = + = ? 15 33 18 36 36 36 The probability of getting an even total when you roll two fair dice is? The probability of getting a prime total when you roll two fair dice is? So why isn’t:

35. The OR Rule People often use the fact that P(A OR B) = P(A) + P(B) But this is only true if the outcomes do not overlap, Outcomes that do not overlap are called MUTUALLY EXCLUSIVE If they do overlap then this rule is no good because you count the overlap twice – its better to count!

36. I roll a red die and a black die. Draw a possibility space diagram to show the total scores for the two dice. Find each probability and decide if the events are mutually exclusive: a) P(A total of 4 OR a total of 6) b) P(A total of 8 OR a total that is prime) c) P(A total that is even OR a total more than 9) d) P(A 4 on the red die OR a 6 on the black die) e) P(A total that is even OR a total that is prime) f) P(A total that is more than 7 OR a 5 on the black die) g) P(A total less than 5 OR a total more than 10)

37. I roll a red die and a black die. Find each probability and decide if the events are mutually exclusive: a) P(A total of 4 OR a total of 6) = 8/36 ME b) P(A total of 8 OR a total that is prime) = 20/36 ME c) P(A total that is even OR a total more than 9) = 20/36 d) P(A 4 on the red die OR a 6 on the black die) = 11/36 e) P(A total that is even OR a total that is prime) = 32/36 ME f) P(A total that is more than 7 OR a 5 on the black die) = 17/36 g) P(A total less than 5 OR a total more than 10) =9/36 ME

38. Lesson Objective Consolidate our ability to find probabilities involving ‘OR’ Consolidate our use of the formula P(A OR B) = P(A) + P(B) for mutually exclusive events and be able to adapt it for events that are not mutually exclusive

39. Eg I roll a fair die labelled 1 to 6. What is the probability that: a) I get an even number? b) I get a square number? c) Either a square number or an even number? d) Is getting a square number a mutually exclusive event to getting an even number? Eg: 1/3 of the students in a room are in Year 7 ¼ of the students in a room are in Year 8 If I pick a random student from the room what is the probability that they are either from Year 7 or Year 8?

40. I roll a red die and a black die. Draw a possibility space diagram to show the total scores for the two dice. Find each probability and decide if the events are mutually exclusive: a) P(A total of 4 OR a total of 6) b) P(A total of 8 OR a total that is prime) c) P(A total that is even OR a total more than 9) d) P(A 4 on the red die OR a 6 on the black die) e) P(A total that is even OR a total that is prime) f) P(A total that is more than 7 OR a 5 on the black die) g) P(A total less than 5 OR a total more than 10)

41. I roll a red die and a black die. Find each probability and decide if the events are mutually exclusive: a) P(A total of 4 OR a total of 6) = 8/36 ME b) P(A total of 8 OR a total that is prime) = 20/36 ME c) P(A total that is even OR a total more than 9) = 20/36 d) P(A 4 on the red die OR a 6 on the black die) = 11/36 e) P(A total that is even OR a total that is prime) = 32/36 f) P(A total that is more than 7 OR a 5 on the black die) = 17/36 g) P(A total less than 5 OR a total more than 10) =9/36 ME

42. 1) The probability that I pick a red sweet from a bag is 0.3. The probability that I pick a yellow sweet is 0.4. What is the probability that a randomly chosen sweet is either red of yellow? 2) Seniors are competitors who are at least 65. Adults are competitors between the ages of 16 and 21 inclusive. Juniors are competitors under the age of 16. 1/5 of the competitors are seniors and 1/3 are juniors. If I randomly pick a competitor what is the probability that they are in the adult category? 3) The probability that it rains on any given day is 0.2. The probability that it is more than 12oC on any given day is 0.4. Why is the probability that it either rains or is warmer than 12oC on any given day not 0.6? 4) In a class of students 1/3 have blue eyes, ¼ have black hair and 1/8 have blue eyes and black hair. If I randomly pick a student at random, what is the probability that they have either blue eyes or black hair?

43. 4) Mr B has a pack of cards, labelled from 1 to 50 If a student randomly picks a card, what is the probability : a) that it is even? b) that it is a multiple of 3? c) that it is either even or a multiple of 3? 5) Mr B has a pack of cards, labelled from 1 to 100 If a student randomly picks a card, what is the probability : a) that it is a multiple of 5? b) that it is a multiple of 7? c) that it is either a multiple of 5 or a multiple of 7? 6) In a class there are 30 students 10 have no pets. The rest of the students either have a cat or a dog or both. 15 students say that they have a dog and 11 students say that they have a cat. What is the probability that a randomly chose student from the class has: a) a pet b) a pet dog c) both a pet and a cat d) either a pet cat or a pet dog or both

44. 7) In a sixth form of 200 students, 60 do maths A level and 45 do Biology A level. 12 study both Maths and Biology. If a randomly chosen person from the sixth form is chosen what is the probability that: a) They study both Maths and Biology b) They study only maths c) Are studying Maths and studying Biology mutually exclusive? 8) Mr B has a pack of cards, labelled from 1 to 100 If a student randomly picks a card, what is the probability : a) that it is a multiple of 3? b) that it is a multiple of 5? c) that it is either a multiple of 5 or a multiple of 7 but not an even number? 9) Make up a question of your own where the probabilities are not mutually exclusive and another question where the probabilities are mutually exclusive.

47. Socks! Four pairs of socks are jumbled up in a drawer. If you put your hand in without looking, how many socks must you take out to be certain of getting a matching pair? What if there were 5 pairs? …. 6? Generalise!

48. Lesson Objective:Be able to find the probability of one thing being followed by another. Begin to understand the difference between P(A followed by B) and P(A and B) Notice: Names and events used in this lesson are random and have no basis in fact. Any resemblance made to actual persons, living or dead, in the class room or outside are purely coincidental.

49. A student in year 10 has 7 pairs of socks. However, being disorganised they simply grab a pair of random socks out of the draw at the start of each day. Unfortunately, they never get round to washing their dirty socks and simply return them to the draw at the end of each day. The draw contains 5 pairs of black socks and 2 pairs of white (?!?) socks. • a) What is the probability that the student wears white socks • on any particular day? • What is the probability that the student wears black socks for two consecutive days? • c) What is the probability that the student wears black socks on Monday followed by white socks on Tuesday, followed by black socks on Wednesday?

50. In general: P(A followed by B followed by C …..) = P(A) × P(B) × P(C) ………… Eg On any given day the probability that a bus is late is 1/3 a) Find the probability that the bus is late two days running. b) Find the probability that the bus is late for three consecutive days. c) Find the probability that from Monday to Friday the bus is late on just Wednesday. d) Find the probability that the bus is not late on the first of three consecutive days, but is late on the other two.