PROBABILITY. Chapter 3

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1 PROBABILITY Chapter 3

2 IN THIS UNIT STUDENTS WILL: Solve contextual problems involving odds and probability. Determine probability using counting methods: Fundamental Counting Principle, Permutations, and Combinations. Use set theory to develop formulas for probabilities of events that are mutually exclusive and not mutually exclusive.

3 3.1 EXPLORING PROBABILITY Chapter 3

4 PROBABILIT Y How likely something is to happen. For example Today there is a chance of rain. The odds of winning the 6-49 lottery are million to one. What are some examples you can think of?

5 PROBABILIT Y How likely something is to happen. If something has a low probability, it is unlikely to happen. If something has a high probability, it is likely to happen.

6 Probabilities can be written as: Decimals from 0 to 1 Ex: Percents from 0% to 100% Ex:

7 PROBABILITY If an event is certain to happen, then the probability of the event is 1 or 100%. EX: If an event will NEVER happen, then the probability of the event is 0 or 0%. EX: If an event is just as likely to happen as to not happen, then the probability of the event is ½, 0.5 or 50%. EX:

8 PROBABILITY SCALE Impossible Unlikely Equal Chances Likely Certain % 50% 100% ½

9 PROBABILITY When a meteorologist states that the chance of rain is 50%, the meteorologist is saying that it is equally likely to rain or not to rain. If the chance of rain rises to 80%, it is more likely to rain. There is a 20% chance that it will not rain.

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11 TOSSING A COIN When a coin is tossed, there are two possible outcomes: heads (H) or tails (T) We say that the probability of the coin landing H is ½. And the probability of the coin landing T is ½.

12 THROWING DICE When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6. The probability of any one of them is 1/6.

13 Probability is always between 0 and 1

14 PROBABILIT Y SCALES Impossible Unlikely Even Chance Likely Certain Place the number in the appropriate location on the scale 1. The sun will rise tomorrow. 2. You will live to be You spin a coin and get a head. 4. You will see a bus on your way home. 5. You will see an ambulance on your way home.

15 PROBABILITY IS JUST A GUIDE Probability does not tell us exactly what will happen, it is just a guide Example: toss a coin 100 times, how many Heads will come up? Probability says that heads have a ½ chance, so we can expect 50 Heads. But when we actually try it we might get 48 heads, or 55 heads... or anything really, but in most cases it will be a number near 50.

16 PROBABILIT Y What are some examples of fair games? A fair game is a game in which all players are equally likely to win. Paper, Rock, Scissors Poker Dice Bingo

17 CALCULATING PROBABILIT Y In general: Probability of an event happening = Number of ways it can happen Total number of outcomes Example: 1. There are 8 marbles in a bag: 4 are blue, 3 are green and 1 is red. What is the probability that a green marble gets picked? Number of ways it can happen: Total number of outcomes: So the probability =

18 EXAMPLES: 2. What is the probability of drawing a king from a deck of cards? 3.What is the possibility of it raining pigs today? 4. What are the chances of rolling an even number on a die?

19 SOME WORDS HAVE SPECIAL MEANING IN PROBABILITY: Experiment or Trial: an action where the result is uncertain. Tossing a coin, throwing dice, seeing what pizza people choose are all examples of experiments. Sample Space: all the possible outcomes of an experiment Similar to the idea of a universal set. Example: What is the sample space for choosing a card from a deck? There are 52 cards in a deck So the Sample Space is all 52 possible cards

20 PG. 141, #1-4 Independent Practice

21 3.2 PROBABILITY AND ODDS Chapter 3

22 PROBABILIT Y SECTION 3.2 The experimental probability of event A, P(A), is defined as the number of times that event A actually occurred, n(a), over the number of trials, n(t). P(A) = n(a) n(t ) The theoretical probability of event A, P(A), is defined as the number of favourable outcomes for event A, n(a), over the total number of outcomes, n(s). P(A) = n(a) n(s) Again this notation was used in set theory. n(s) is the number of elements in the sample space

23 DEFINITIONS Probability: The ratio of favourable outcomes to the total possible outcomes. Example: What is the probability that the spinner lands on an odd number? This probability can be written as a ratio: Or More commonly, it can be written as a fraction: Or It can be written as a percentage. OR it can be written as ( part: whole ) Odds: the ratio of favourable outcomes to unfavourable outcomes. Odds are always written as a ratio: ( part: part ) part whole

24 NOTATION FOR ODDS When we solve problems involving odds, we examine the number of favourable outcomes and the number of unfavourable outcomes. We use the following notation: o n( A) represents the number of favourable outcomes. o n( A') represents the number of unfavourable outcomes. o We read this as not ( A) or the complement of ( A)

25 Odds in favor of an event would be: favourable : not favourable n( A) : n( A') Odds against an event would be: not favourable : favourable n( A') : n( A) And remember: favourable + not favourable = SampleSpace n(a) +n(a')= Total outcomes AND P( A) P( A') 1

26 EXAMPLE 1. USING A FAIR SIX-SIDED DIE, CALCULATE EACH OF THE FOLLOWING: A. The probability of rolling a 2. B. The odds in favour of rolling a 2. C. The odds against rolling a 2.

27 EXAMPLE 2 Suppose that, at the beginning of a regular CFL season, the Saskatchewan Roughriders are given a 25% chance of winning the Grey Cup. a) What is the probability that the event will occur as a fraction? b) Describe the complement of this event as a percentage? c) Express the probability of the complement of this event as a fraction. d) Write the odds in favour of the Roughriders winning the Grey Cup. e) Write the odds against the Roughriders winning the Grey Cup.

28 EXAMPLE 3 Bailey holds all the hearts from a standard deck of 52 playing cards. He asks Morgan to choose a single card without looking. Determine the odds in favour of Morgan choosing a face card. Consider the set of possible heart cards: H = {A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K} What is the set, F, of face cards? How many cards are there? F = {J, Q, K} 3 cards What is the complement of the set, F? How many cards are there? F = {A, 2, 3, 4, 5, 6, 7, 8, 9, 10} 10 cards Odds in Favour n(f) : n(f ) Odds in Favour 3 : 10

29 EXAMPLE 4 Research shows that the probability of an expectant mother, selected at random, having twins is. A. What are the odds in favour of an expectant mother having twins? B. What are the odds against an expectant mother having twins? A. B. The odds against having twins is P(not twins) : P(twins) The odds in favour of having twins is P(twins) : P(not twins)

30 EXAMPLE 5

31 EXAMPLE 6. A computer randomly selects a student s name from the school s database to award a $500 gift certificate. The odds against the selected student being male are 57 : 43. Determine the probability that the randomly selected student will be male. What is the number of males in the database? 57 : 43 represents the n(f) : n(m) the number of males is 43 What is the number of females in the database? 57 So what s the total number of outcomes? = 100 The probability of randomly selected university student being male is 43%.

32 EXAMPLE 7. Suppose that the odds in favour of an event are 5 : 3. Is the probability that the event will happen greater or less than 50%?

33 EXAMPLE 8. A group of Grade 12 students are holding a charity carnival to support a local animal shelter. The students have created a dice game that they call Bim and a card game that they call Zap. The odds against winning Bim are 5 : 2, and the odds against winning Zap are 7 : 3. Which game should Madison play? What s the total number of outcomes for Bim? = 7 What s the total number of winning outcomes for Bim? 2 So, the probability of winning Bim is: P(winning Bim) = 2/7 P(winning Bim) = What s the total number of outcomes for Zap? = 10 What s the total number of winning outcomes for Zap? 3 So, the probability of winning Zap is: P(winning Zap) = 3/10 P(winning Zap) = 0.3 There is a greater chance of winning Zap, so she should play that.

34 PG , #1, 4, 5, 8, 10, 14, 17. Independent Practice

35 PROBABILIT Y.HTML

36 3.2 CHECK-UP. 1. Probability or Odds? A. The chances of rolling a 5 on a fair six-sided dice is 1 6 B. The chances of drawing an Ace from a standard 52 -card deck is 1:12 C. The chance of flipping a coin and it landing on heads is 50%.

37 2. WHICH OF THE FOLLOWING ODDS FOR AND PROBABILITY STATEMENTS ARE CORRECT? WHY OR WHY NOT? Odds for Example 1 1:3 Example 2 4:5 Example 3 4:6 Probability

38 REVIEW CONTINUED 3. The odds of winning a contest are 5:9. A. What does the sum of the parts represent? B. What is the probability of winning the contest? C. What is the probability of losing the contest?

39 4. The probability of you passing the next Math test is 75%. What are the odds of you passing?

40 5 A jar contains three red marbles and some green marbles. The odds are 3:1 that a randomly chosen marble is green. How many green marbles are in the jar?

41 3.3 PROBABILITY USING COUNTING Chapter 3 METHODS

42 WHEN PULLING A SINGLE ITEM FROM A SET, WE USE BASIC PROBABILIT Y. 1. A bag contains three red balls and four green balls. One ball is pulled out. What is the probability of: A. Pulling out a red ball? B. Pulling out a green ball? C. Pulling out a blue ball?

43 2. A LETTER IS SELECTED FROM THE WORD STATISTICS. WHAT IS THE PROBABILITY THE LETTER IS: A. An S or a T? B. Not a vowel?

44 PROBABILITY INVOLVING FUNDAMENTAL COUNTING PRINCIPAL 1. A 5-digit PIN can begin with any digit (except zero) and the remaining digits have no restriction. If repeated digits are allowed, find the probability of the PIN beginning with a 7 and ending with an 8. Number of favorable outcomes = = 1000 Total number of outcomes = = Probability =

45 2. THREE DIFFERENT DVDS AND THEIR CORRESPONDING DVD CASES ARE RANDOMLY STREWN ABOUT ON A SHELF. IF A YOUNG CHILD PUTS THE DVDS IN THE CASES AT RANDOM, DETERMINE THE PROBABILIT Y OF CORRECTLY MATCHING ALL DVDS AND CASES. Favorable outcomes: DVD A DVD B DVD C Total outcomes: DVD A DVD B DVD C Probability of correctly matching all DVDs and their cases: 1 6 or or 16. 7%

46 3. EACH LETTER OF THE WORD PENCIL IS WRITTEN ON A SEPARATE CARD. WHAT IS THE PROBABILIT Y THAT IF 2 CARDS ARE SELECTED, THEY WILL BOTH BE VOWELS?

47 4. M S. H Y NES IS ARRANGING 5 B O O K S O N A SHELF - LO RD O F T H E FLIE S, ANIMAL FA RM, NIGHT, RO M EO & JULIET, A ND JULIUS CAESAR. A)WHAT IS THE PRO B A B ILIT Y T H AT LO RD O F THE FLIES AND NIGHT ARE NEXT TO EAC H OTHER?

48 B) WHAT IS THE PROBABILIT Y THAT LORD OF THE FLIES AND NIGHT ARE NOT NEXT TO EACH OTHER ON THE SHELF?

49 INDEPENDENT PRACTICE Page #1,5,9,12,15

50 WARM UP (FROM SAMPLE PUBLIC EXAM) If a 5-digit number is generated at random from the digits 2, 3, 4, 5 and 8 (with no repetition), what is the probability that it will be an odd number?

51 PROBABILITY INVOLVING PERMUTATIONS AND COMBINATIONS Permutations refers to the numbers of ways a group of objects can be arranged, such that the ORDER IS IMPORTANT. n Pr n! ( n r)! where n is the number of choices and r is the number of objects to permute Combinations refers to the numbers of ways a group of objects can be arranged, such that the ORDER IS NOT IMPORTANT. n Cr n! ( n r)! r! where n is the number of choices, r is the number of objects chosen and n r is the number of objects NOT chosen

52 1. USING A STANDARD DECK OF 52 CARDS, AND SELECTING 5 CARDS, WHAT IS THE PROBABILIT Y OF: A. Getting a flush? (all five cards are the same suit) total hands = C 2,598, hands getting a flush = 13C5 13C5 13C5 13C hands ( Spades ) ( Hearts ) ( Diamonds ) ( C lub s) Probability of getting a flush = , 598, %.

53 B. HAVING AT LEAST 3 SPADES? With 3 spades or with 4 spades or with 5 spades 13 spades 39 other spades 39 other spades 39 other 5 0 C C C C C C Probability of at least 3 spades= % 2,598,960

54 C) WHAT ARE THE ODDS IN FAVOUR OF GETTING 4 OF A KIND? How four of a kinds are there? C C How many ways can you fill in the remaining card? How do we find the total number of 4 of a kind? How do we find the total number of hands that are not 4 of a kind? Subtract the total number by the number of 3 of a kinds 2,598,960 Odds in favour of getting 4 of a kind = : :

55 2. THERE ARE 12 MALE ATHLETES AND 14 FEMALE ATHLETES COMPETING IN A MARATHON. FIND THE PROBABILIT Y OF THE TOP THREE PRIZES BEING AWARDED TO: A. All males P P B. All females P P C. All males or all females P P 26 3 P

56 3. A BOOKCASE CONTAINS 6 DIFFERENT MATH BOOKS AND 1 2 DIFFERENT PHYSICS BOOKS. IF A STUDENT RANDOMLY SELECTS T WO BOOKS, A) DETERMINE THE PROBABILIT Y THEY ARE BOTH MATH OR BOTH PHYSICS BOOKS. C C C C 18 2 C B) What are the odds of the 2 books NOT being 2 Math or 2 Physics books?

57 4. A JAR CONTAINS 5 ORANGE, 3 PURPLE, 7 BLUE, AND 5 GREEN CANDIES. DETERMINE THE PROBABILIT Y THAT A HANDFUL OF FOUR CANDIES CONTAINS ONE OF EACH COLOUR. C C C C C %

58 5. A COMMITTEE OF 4 PEOPLE IS TO BE FORMED FROM A POOL OF 8 PEOPLE. WHAT ARE THE ODDS AGAINST JAMES IS ON THE COMMITTEE? # of ways James is not on committee: # of was James is on committee.

59 6. LOTTO 7-45 HAS 45 NUMBERS YOU CAN CHOOSE FROM, AND YOU MUST PICK 7. IF A FREE TICKET IS GIVEN OUT BY MATCHING 2 CORRECT NUMBERS, WHAT IS THE PROBABILIT Y OF GETTING THIS PARTICULAR PRIZE? P(2 correct # sand 5 others ) C C 45 7 C 10,540,782 45,379,

60 7. Justin, Ethan, and Liam are competing with seven other boys to be on their school s cross-country team. All the boys have an equal chance of winning the trial race. Determine the probability that Justin, EXAMPLE Ethan, and Liam 7will place first, second, and third, in any order. How many ways can they all place in the top three? P P How many ways can any of the 10 runners place first, second or third? There are 6 favourable outcomes. There are 720 possible outcomes. Note: This problem could also be done with combinations: C 3 3 C 10 3

61 8. About 20 years after they graduated from high school, Blake, Mario, and Simon met in a mall. Blake had two daughters with him, and he said he had three other children at home. Determine the probability that at least one of Blake s children is a boy. Use Indirect reasoning What are the possibilities for each of the three children at home? How many outcomes are there for where each child is a girl? Only 1 So, in how many ways is there at least one boy? P(at least one boy) =

62 9. Jacob hosts a morning radio show in Saskatoon. To advertise his show, he is holding a contest at a local mall. He spells out SASKATCHEWAN with letter tiles. Then he turns the tiles face down and mixes them up. He asks Sally to arrange the tiles in a row and turn them face up. If the row of tiles spells SASKATCHEWAN, Sally will win a new car. Determine the probability that Sally will win the car? In how many ways can Sally win the car? One What is the total number of possible arrangements of the tiles? 12! 2!3! What is the probability that Sally wins a car ( express answer as a ratio)?

63 INDEPENDENT PRACTICE Pages # 2, 3, 4, 6, 8, 10, 11, 15, 16, 17

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67 3.4 MUTUALLY EXCLUSIVE EVENTS Chapter 3

68 MUTUALLY EXCLUSIVE EVENTS Mutually exclusive: can't happen at the same time. Examples: Turning left and turning right are Mutually Exclusive (you can't do both at the same time) Tossing a coin: Heads and Tails are Mutually Exclusive Cards: Kings and Aces are Mutually Exclusive

69 WHAT IS NOT MUTUALLY EXCLUSIVE: Turning left and scratching your head can happen at the same time Kings and Hearts, because you can have a King of Hearts! Like here: Aces and Kings are Mutually Exclusive (they never intersect) Hearts and Kings are not Mutually Exclusive (intersection is King of Hearts)

70 MUTUALLY EXCLUSIVE EVENTS When two events (call them "A" and "B") are Mutually Exclusive, it is impossible for them to happen together. That is, The sets have no common elements. P(A and B) = 0 The probability of A and B happening together is 0 (impossible)

71 BUT THE PROBABILITY OF A OR B IS THE SUM OF THE INDIVIDUAL PROBABILITIES: P(A or B) = P(A) + P(B) "The probability of A or B equals the probability of A plus the probability of B"

72 EXAMPLE 1: IN A STANDARD DECK OF 52 CARDS, WHAT IS THE PROBABILITY OF: A) Getting a King? B) Getting an Ace? 4 P( King) 52 4 P( Ace) C) Of a card being a King and an Ace? D) Of a card being a King or an Ace? P( King and Ace) P( King or Ace)

73 SPECIAL NOTATION Instead of "and" you will often see the symbol (which is the "Intersection" symbol) Instead of "or" you will often see the symbol (the "Union" symbol)

74 EXAMPLE 2: SCORING GOALS If the probability of scoring no goals (Event "A") is 20% and scoring exactly 1 goal (Event "B") is 15%, then: A) The probability of scoring no goals and 1 goal is P(A B) = 0 B) The probability of scoring no goals or 1 goal is P(A B) = 20% + 15% = 35%

75 REMEMBER! To help you remember, think: "Or has more... than And is like a cup which holds more than

76 NOT MUTUALLY EXCLUSIVE EVENTS (OVERLAPPING SETS) Example 1: Hearts and/or Kings P(Hearts and Kings) is the King of Hearts P(Hearts or Kings) is: all the Hearts (13 of them) all the Kings (4 of them) But that counts the King of Hearts twice!

77 SO WE CORRECT OUR ANSWER, BY SUBTRACTING THE EXTRA "AND" PART: 16 Cards = 13 Hearts + 4 Kings - the 1 extra King of Hearts

78 AS A FORMULA THIS IS: P(A or B) = P(A) + P(B) - P(A and B) "The probability of A or B equals the probability of A plus the probability of B minus the probability of A and B Here is the same formula, but using and : P(A B) = P(A) + P(B) - P(A B)

79 EXAMPLE PEOPLE STUDY FRENCH, 21 STUDY SPANISH. IF THERE ARE 30 PEOPLE SURVEYED AND EVERYONE STUDIES AT LEAST ONE OF THESE LANGUAGES, HOW MANY STUDY BOTH FRENCH AND SPANISH? Let's say b is how many study both languages: people studying French Only must be 16-b people studying Spanish Only must be 21-b And we know there are 30 people, so: (16-b) + b + (21-b) = b = 30 b = 7

80 AND NOW WE CAN PUT IN THE CORRECT NUMBERS: Now we can find: P(French) = P(Spanish) = P(French Only) = P(Spanish Only) = P(French or Spanish) = P(French and Spanish) =

81 Lastly, let's check with our formula: P(FS) = P(F) + P(S) - P(FS)

82 SUMMARY: Mutually Exclusive The probability of A and B together is impossible: P(A B) = 0 The probability of A or B is the sum of A and B: P(A B) = P(A) + P(B) Not Mutually Exclusive The probability of A or B is the sum of A and B minus A and B: P(A B) = P(A) + P(B) - P(A B)

83 1. CLASSIFY THE EVENTS IN EACH EXPERIMENT AS BEING EITHER MUTUALLY EXCLUSIVE OR NON-MUTUALLY EXCLUSIVE. A) The experiment is rolling a die. The first event is rolling an even number and the second event is rolling a prime number. B) The experiment is playing a game of hockey. The first event is that your team scores a goal, and the second event is that your team wins the game. C) The experiment is selecting a gift. The first event is that the gift is edible and the second event is that the gift is an iphone.

84 EXAMPLE 2 A school newspaper published the results of a recent survey. a) Are skipping breakfast and skipping lunch mutually exclusive events? b) Determine the probability that a randomly selected students skips breakfast but not lunch. c) Determine the probability that a randomly selected student skips at least one of breakfast or lunch.

85 EXAMPLE 3 Reid s mother buys a new washer and dryer set for $2500 with a 1-year warranty. She can buy a 3-year extended warranty for $450. Reid researches the repair statistics for this washer and dryer set and finds the data in the table below. Should Reid s mother buy the extended warranty?

86 EXAMPLE 4 In a board game, each player rolls two dice. If a player rolls a sum that is greater than 8 or a multiple of 5, the player gets a bonus of 100 points. Determine the probability that you will receive a bonus of 100 points on your next roll. Possible Sums When a Pair of Dice is Rolled

87 EXAMPLE 5 A car manufacturer keeps a database of all the cars that are available for sale at all the dealerships in Western Canada. For model A, the database reports that 43% have heated leather seats, 36% have a sunroof, and 49% have neither. Determine the probability of a model A car at a dealership having both heated seats and a sunroof.

88 PG , # 2, 3, 5, 7, 8, 14, 17 Independent Practice

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90 SECTION 3.4 CHECK-UP

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92 MUTUALLY EXCLUSIVE EVENTS 1. Using the Venn Diagram below, determine P(AB) A B 25

93 . NOT MUTUALLY EXCLUSIVE EVENTS 2.The probability that Dana will make the hockey team is. The probability that she will make the swimming team is the probability of Dana making both teams is the probability that she will make: , determine. If A) At least one of the teams B) Neither team

94 . This problem can also be completed using a Venn diagram 2.The probability that Dana will make the hockey team is. The probability that she will make the swimming team is the probability of Dana making both teams is the probability that she will make: , determine. If A) At least one of the teams P(H S) = P(H) + P(S) - P(H S) H S B) Neither team Hint: Write all fractions with the same denominator OR :Write all fractions as decimals or percents

95 . NOT MUTUALLY EXCLUSIVE EVENTS 2.The probability that Dana will make the hockey team is. The probability that she will make the swimming team is the probability of Dana making both teams is the probability that she will make: , determine. If A) At least one of the teams H S B) Neither team

96 3. The probability that the Toronto Maple Leafs will win their next game is 0.5. The probability that the Montreal Canadiens will win their next game is 0.7. The probability that they will both win is Determine the probability that one or the other will win their next game (assume they don t play each other). Include a Venn Diagram in your solution. Leafs Canadiens

97 3. The probability that the Toronto Maple Leafs will win their next game is 0.5. The probability that the Montreal Canadiens will win their next game is 0.7. The probability that they will both win is Determine the probability that one or the other will win their next game (assume they don t play each other). Include a Venn Diagram in your solution. Leafs Canadiens

98 PROBABILITIES OF COMPLEMENTARY EVENTS Suppose you are rolling a fair 6-sided dice. A. What is the probability of rolling a 2? B. What is the probability of NOT rolling a 2? C. What do you notice about the two probabilities?

99 COMPLEMENTARY EVENTS HAVE PROBABILITIES THAT ADD UP TO GIVE 1. That is, P(A) + P(A') = 1 Thus, we could find the probability of the complement of event A using... P(A') = 1 - P(A)

100 5. P(A) = 0.6, P(B) = 0.7, P(AB)=0.15, DETERMINE P((AB))

101 6.A and B are mutually exclusive events. The probability that either A or B will occur [ P(AB)] is 56%. If the probability of A occurring [P(A)] is 17%, what is the probability of B not occurring, [P(B)]? (A) 27% (B) 39% (C) 61% (D) 73%

102 3.6 INDEPENDENT EVENTS Chapter 3

103 INDEPENDENT EVENTS To determine whether two events are dependent or independent, we determine whether one event will af fect the probable outcome of another. If one event does NOT affect the other, then the two events are said to be independent of each other. If one event does affect the other, then they are said to be dependent, and we use something called conditional probability.

104 DETERMINE WHETHER THE FOLLOWING EVENTS ARE DEPENDENT OR INDEPENDENT. Example 1: Event A: drawing a queen from a standard deck of cards Event B: drawing a king from the remaining cards in the same deck (without replacement) Dependent Example 2: Event A: rolling a 5 on a die Event B: rolling a 3 on the same die Independent

105 CLASSIFY THE FOLLOWING EVENTS AS EITHER INDEPENDENT OR DEPENDENT AND EXPLAIN WHY. (i) The experiment is rolling a die and flipping a coin. The first is rolling a six and the second event is obtaining tails. Independent. Rolling a die does not affect flipping a coin (ii) The experiment is rolling a pair of dice. The first event is rolling an odd number on one die and the second event is rolling an even number on the other die. Independent. Neither die as an affect on the other.

106 (iii) The experiment is dealing 5 cards from a standard deck. The first event is that the first card dealt is a spade, the second event is that the second card is a spade, the third event is that the third card is a spade and so on. Dependent. The second card depends on what the first card is. (iv) The experiment is to sample two members of a family, a mother and her child. The first event is that the mother has blond hair and the second event is that the child has blond hair. Dependent. Genetics

107 FORMULA FOR INDEPENDENT EVENTS When two events A and B are independent of each other, we can find the probability of A and B happening by using... P( Aand B) P( A) P( B)

108 EXAMPLES 1. A dresser drawer contains one pair of socks with each of the following colors: blue, brown, red, white and black. Each pair is folded together in a matching set. You reach into the sock drawer and choose a pair of socks without looking. You replace this pair and then choose another pair of socks. What is the probability that you will choose the red pair of socks both times? P(red and red) = P(red) P(red)

109 EXAMPLE 2: DETERMINE THE PROBABILIT Y OF ROLLING A 3 ON A DICE AND TOSSING HEADS ON A COIN.

110 3. A jar contains 3 red, 5 green, 2 blue and 6 yellow marbles. A marble is chosen at random from the jar. After replacing it, a second marble is chosen. What is the probability of choosing a green and then a yellow marble?

111 4. In a playoff series, the probability that Team A wins over Team B is 1/2, and the probability that Team C wins over Team D is 3/5. If the probabilities are independent events, then what is the probability that Team A wins and Team C loses?

112 5. CONSIDER THE EXPERIMENT OF TOSSING A COIN T WICE WHERE WE ARE INTERESTED IN THE NUMBER OF HEADS. A) What is the probability of getting 2 heads? P(H and H) P( H) P( H)

113 5. CONSIDER THE EXPERIMENT OF TOSSING A COIN T WICE WHERE WE ARE INTERESTED IN THE NUMBER OF HEADS. B) What is the probability of getting at least 1 tail? When doing multiple outcomes it can be convenient to determine probability of two (or more) independent events using a tree diagram. What is the probability getting at least 1 tail? What is the total probability?

114 PROBABILIT Y TREE DIAGRAMS Lets first look at probability tree diagrams as puzzles. In other words, were going to play with the tool, without even touching on the theory or concepts. Which means if you can follow these puzzle rules, you can do the probability tree diagrams. And when (or if) you need to use them for calculating probabilities, you ll know how!

115 HERE ARE THE RULES FOR THE PROBABILIT Y TREE DIAGRAM PUZZLE: 1. All of the branches coming from a particular node have to add up to one. 2. All of the leaves on the very end have to add up to one. 3. Each leaf at the very end is calculated by multiplying all numbers down the branches leading to that leaf.

116 COMPLETE THIS ONE! Start 0.25 A B 0.3 C D C D 0.6

117 1. PRACTICE:

118 2.

119 FUN ONE!

120 EXAMPLE 1: DETERMINE THE PROBABILIT Y OF ROLLING A PRIME NUMBER ON A DIE AND TOSSING HEADS ON A COIN. A) USE a tree diagram to solve this problem B) What other information can you determine using the tree diagram? You can answer any question about rolling a die and flipping a coin

121 C) What is the probability of getting a 6 and a head? D) What is the probability of getting either a 6 OR a head (but not both)? E) What is the probability of getting neither a 6 nor a head?

122

123 EXAMPLE 2 A trick coin is designed so that the probability of it landing heads is 5/8. If the coin is flipped until exactly 2 tails appear, what is the probability that it will take exactly three flips to get the second tail?

124 EXAMPLE 3 Ryan and Jessica are playing a die and coin game. Each turn consists of rolling a regular die and tossing a coin. Points are awarded for rolling a 6 on the die and/or tossing heads with the coin: 1 point for either outcome 3 points for both outcomes 0 points for neither outcome Players alternate turns. The first player who gets 10 points wins. Determine the probability that Ryan will get 1, 3, or 0 points on his first turn.

125 EXAMPLE 3 Ryan and Jessica are playing a die and coin game. Each turn consists of rolling a regular die and tossing a coin. Points are awarded for rolling a 6 on the die and/or tossing heads with the coin: 1 point for either outcome 3 points for both outcomes 0 points for neither outcome Players alternate turns. The first player who gets 10 points wins. Determine the probability that Ryan will get 1, 3, or 0 points on his first turn.

126 EXAMPLE 4 All 1000 tickets for a charity raffle have been sold and placed in a drum. There will be two draws. The first draw will be for the grand prize, and the second draw will be for the consolation prize. After each draw, the winning ticket will be returned to the drum so that it might be drawn again. Max has bought five tickets. Determine the probability, to a tenth of a percent, that he will win at least one prize.

127

128 EXAMPLE 4 All 1000 tickets for a charity raffle have been sold and placed in a drum. There will be two draws. The first draw will be for the grand prize, and the second draw will be for the consolation prize. After each draw, the winning ticket will be returned to the drum so that it might be drawn again. Max has bought five tickets. Determine the probability, to a tenth of a percent, that he will win at least one prize.

129

130 PG , #1, 3, 5, 7, 8, 11, 18 Independent Practice

131 3.5 CONDITIONAL PROBABILITY Chapter 3

132 MARBLES IN A BAG 2 blue and 3 red marbles are in a bag. What is the probability of getting a blue marble? What is the probability that the next marble is blue? THAT DEPENDS on whether the first marble was placed back in the bag. If the first marble is put back (replacement) this would be an independent event. Then the probability that the next marble is blue is: And the probability of both marbles drawn are blue is:

133 MARBLES IN A BAG If the marble is not put back (without replacement) after taking one out, the probabilities change! What is the probability of getting a blue marble on the first draw? Then the probability that the next marble is blue is: And the probability of both marbles drawn are blue is: NOTE: If we got a red marble on the first, then the chance of a blue marble next is:

134 Dependent Events: Events whose outcomes are affected by each other. For example, if two marbles are drawn from a bag without replacement, the outcome of the second event depends on the outcome of the first event (the first marble drawn). The probability of an event occurring given that another event has already occurred is called Conditional Probability.

135 The methods that we will use to solve probability problems involving conditional events are very similar to what we used previously. The only difference is in the way that the probability for the second event is calculated. Since the second event depends on the first, the favorable and total outcomes for the second event will need to be adjusted accordingly.

136 1. CARDS ARE DRAWN FROM A STANDARD DECK OF 52 CARDS WITHOUT REPLACEMENT. CALCULATE THE PROBABILIT Y OF OBTAINING: A) A king, given that the first card drawn was i) the Ace of hearts ii) the King of hearts NOTE: The probability of event B given event A is written as: Write each of the above using this notation. P B A

137 FORMULA To find the probability of A and B if the events are dependent, P A B multiply the probability of A and the probability of B given that A occurred. That is, ( ) P A B P A x P B A Note: This formula can be rearranged to calculate P A B P B A PA ( ) P B A

138 1. CARDS ARE DRAWN FROM A STANDARD DECK OF 52 CARDS WITHOUT REPLACEMENT. CALCULATE THE PROBABILIT Y OF OBTAINING: B) A king, then another king P K 2 nd K

139 C) A club, then a heart

140 D) A black card, then a heart, then a diamond

141 2. A JA R CONTA INS B LAC K A ND WHITE M A RBLES. T WO M ARBLES ARE C HOSEN WITHOUT REPLAC EMENT. T H E PRO B A B ILIT Y O F SELECTING A B LAC K M A RBLE AND THEN A WHITE MARBLE IS 0.34, AND THE P RO B A B ILIT Y O F SELECTING A BLAC K M A RBLE ON THE FIRST DRAW IS WHAT IS THE PRO B A B ILIT Y O F : Selecting a white marble on the second draw, given that the first marble drawn was black?

142 3. A BAG CONTAINS 4 RED BALLS AND 3 GREEN BALLS. A BALL IS TAKEN AT RANDOM. A SECOND BALL IS THEN TAKEN FROM THE BAG. (NO REPLACEMENT) A) Are the events independent? Why or why not? B) What is the probability of picking a red ball and then a green ball?

143 ICE CREAM 4. 70% of your friends like Chocolate, and 35% like Chocolate AND like Strawberry. What percent of those who like Chocolate also like Strawberry? P(Strawberr y Chocolate) = P(Chocolate and Strawberr y) / P(Chocolate)

144 SOCCER GAME 5. You are off to soccer, and want to be the Goalkeeper, but that depends who is the Coach today: with Coach Sam the probability of being Goalkeeper is 0.5 with Coach Alex the probability of being Goalkeeper is 0.3 Sam is Coach more often... about 6 out of every 10 games (a probability of 0.6). So, what is the probability you will be a Goalkeeper today? Let's build a tree diagram. First we show the two possible coaches: Sam or Alex: The probability of getting Sam is 0.6, so the probability of Alex must be 0.4 (together the probability is ) 1

145 Now, if you get Sam, there is 0.5 probability of being Goalie (and 0.5 of not being Goalie): If you get Alex, there is 0.3 probability of being Goalie (and 0.7 not): Calculate the overall probabilities. This is done by multiplying each probability along the "branches" of the tree.

146 So, what is the probability you will be a Goalkeeper today? = 0.42 probability of being a Goalkeeper today (That is a 42% chance)

147 Which student correctly identified the probability and why? 6. A H O C K EY TEAM HAS JERSEYS IN 3 DIFFERENT C O LO U RS. T H ERE A RE 4 GREEN, 6 WHITE AND 5 ORANGE JERSEYS IN THE HOCKEY BAG. TO D D AND BLAKE ARE GIVEN A JERSEY AT RANDOM. S T U D ENTS WERE A S K ED TO WRITE A N EXPRESSION REPRESENTING T H E PRO B A B ILIT Y T H AT B OTH JERSEYS ARE THE SAME C O LO U R. Tony Sam Lesley Dana

148 EXAMPLE 7 A computer manufacturer knows that, in a box of 100 chips, 3 will be defective. Jocelyn will draw 2 chips, at random, from a box of 100 chips. Determine the probability that Jocelyn will draw 2 defective chips. Let A represent the event that the first chip I draw will be defective. Let B represent the event that the second chip I draw will be defective. What s the probability that the first chip I draw will be defective? PA ( ) What s the probability that the second chip I draw will be defective? P( B A) 2 99 P( A B) P( A) P( B A)

149 EXAMPLE 8 According to a survey, 91% of Canadians own a cellphone. Of these people, 42% have a smartphone. Determine, to the nearest percent, the probability that any Canadian you met during the month in which the survey was conducted would have a smartphone. Let C represent owning a cellphone. Let S represent owning a smartphone. Smartphones are a subset of cellphones, so the probability of having a smartphone is the same as the probability of both a cellphone and a smartphone.

150 EXAMPLE 9 Hillary is the coach of a junior ultimate team. Based on the team s record, it has a 60% chance of winning on calm days and a 70% chance of winning on windy days. Tomorrow, there is a 40% chance of high winds. There are no ties in ultimate. What is the probability that Hillary s team will win tomorrow? What s the probability of it being windy? Then what s the probability of it being calm? Draw a tree diagram:

151 EXAMPLE 9 Hillary is the coach of a junior ultimate team. Based on the team s record, it has a 60% chance of winning on calm days and a 70% chance of winning on windy days. Tomorrow, there is a 40% chance of high winds. There are no ties in ultimate. What is the probability that Hillary s team will win tomorrow? What s the probability of it being windy? Then what s the probability of it being calm? Draw a tree diagram:

152 PG , #1, 3, 5, 6, 8, 9, 10, 14, 19. Independent practice

153 9. Ian likes to go for daily jogs with his dog, Oliver. If the weather is nice, he is 85% likely to jog for 8 km. If the weather is rainy, he is only 40% likely to jog for 8 km. The weather forecast for tomorrow indicates a 30% chance of rain. Determine the probability that Ian will jog for 8 km.

154 9. Ian likes to go for daily jogs with his dog, Oliver. If the weather is nice, he is 85% likely to jog for 8 km. If the weather is rainy, he is only 40% likely to jog for 8 km. The weather forecast for tomorrow indicates a 30% chance of rain. Determine the probability that Ian will jog for 8 km.

155 8. Anita remembers to set her alarm clock 62% of the time. When she does remember to set her alarm clock, the probability that she will be late for school is When she does not remember to set it, the probability that she will be late for school is Anita was late today. What is the probability that she remembered to set her alarm clock?

156 8. Anita remembers to set her alarm clock 62% of the time. When she does remember to set her alarm clock, the probability that she will be late for school is When she does not remember to set it, the probability that she will be late for school is Anita was late today. What is the probability that she remembered to set her alarm clock?

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