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Math 12 Advanced: An Introduction to Probability – you should probably do this!!! PAGE 300 Section 5.1 Probability and Quantifying the Outcomes P( X ) The notation P( X ) means the probability of event X occurring. Number of Ways of X Occurring Total Number of Possible Outcomes Example: Rolling a six sided die, the sample space denoted S {1,2,3,4,5,6} represents all the possible outcomes. The probability of rolling and even number: P(even) 3 or 0.5 6 The probability of rolling a 1, 2, 3, 4, 5, or 6: P(1,2,3,4,5 or 6) 6 or 1 6 Probability ranges from zero to one! 0 The probability of rolling a 7: P(7) or 0 6 Example: Suppose you pick one card from a well shuffled deck of 52 playing cards. What is the probability of drawing a king? P(King ) What is the probability of not drawing a king? 4 1 or 52 13 P( Not a King ) 48 12 or 52 13 The probability of “drawing a king” and “not drawing a king” are complements of each other! P( X ) 1 P( X ) The complement of X is often written as X . Homework Questions: 1. Given a well shuffled deck of 52 playing cards, you pick one card, determine the probability of the following: a) Drawing a 5 b) Drawing a red 5 c) Drawing a face card d) Drawing the King of Spades e) Not drawing the King of Spades f) Drawing a red card g) Drawing an Ace or Queen h) Not drawing an Ace or Queen i) Drawing a 7, 8 or 9 2. A bag contains three black marbles, six red marbles and 6 white marbles. What is the probability of choosing each of the following: a) a black marble b) a white marble c) a red or a white marble 3. A hockey team has 2 goalies, six defence and nine forwards. Suppose that one of them is drafted. Calculate each probability: a) The draftee is a goalie. b) The draftee is not a forward. 4. Given 2 fair die to roll, determine the sample space of possible sums by completing the chart below. Then determine the probability of the following: Die 1 Die 2 1 1 2 3 4 5 6 a) Rolling a sum of 7 b) Rolling a sum of 11 2 c) Not rolling a sum of 11 3 d) Rolling a sum of 1 4 e) Rolling an even sum 5 f) Rolling 2 sixes 6 From page 302 in you text book: 5. text page 302 #5 6. text page 302 #6 7. a) Explain the difference between theoretical probability and experimental probability. b) Give an example of a situation in which you would rely on experimental probability rather than on theoretical probability 8. In theory, what is the P(passing a test)? Explain your answer. 9. If the P(passing a test) = 7 8 and 32 students write the test, how many should pass? Would this be an example of theoretical or experimental probability? Explain. PAGE 307 Section 5.2 Counting and Probability Consider 3 cars in a race – a dodge, a ford and a Honda. How many different ways can they finish 1st and 2nd? Consider the chart below: First Dodge Dodge Ford Ford Honda Honda Second Ford Honda Dodge Honda Ford Dodge Consider the Tree Diagram below: Dodge DF DH FD FH HF HD Ford Honda Ford (DF) Honda (DH) Dodge (FD) Honda (FH) Ford (HF) Dodge (HD) How many different possibilities are there? Consider the following situation: Lori is going to the show. She will go with either Lynne or Chris, they will see a comedy, a horror or a drama and they will order licorice or popcorn or nachos. How many possibilities must she consider?×× Licorice Popcorn Comedy Nachos Lynne Chris One Possibility: Lynne, Comedy and Licorice Horror Licorice Popcorn Nachos Drama Licorice Popcorn Nachos Comedy Licorice Popcorn Nachos Would this method have worked with the car race above? Horror Licorice Popcorn Nachos Drama Licorice Popcorn Nachos This is called the Fundamental Counting Principle! So how many possibilities does Lori have to consider in total? How could we have achieved the answer of “18” more efficiently? × (number of first choices) × (number of second choices) = (number of third choices) (total number of choices) Homework Questions: page 308 #1 – 9 Applying the Fundamental Counting Principle to Probability Read through Focus B on page 309: then do questions 10 – 12 on page 310 (should be doing some multiplying of probabilities) PAGE 310: Dependent and Independent Events Dependent event - Event A is dependent of event B if the result of one does affect the result of the other. Example: A jar contains three green marbles, four purple marbles and a black marble. Given a green marble (event A) is drawn and not put back in the bag (not replaced) what is the probability of drawing a purple marble (event B)? Answer: PPurple 4 or 0.57 7 There are 4 purple marbles out of now only 7 total. Independent event - Event A is independent of event B if the result of one does not affect the results of the other. Example: A jar contains three green marbles, four purple marbles and a black marble. Given a green marble (event A) is drawn and the put back in the bag (replaced) what is the probability of drawing a purple marble (Event B)? Answer: PPurple 4 1 or 0.5 8 2 There are 4 purple marbles out of 8 total. When there is replacement, the event is considered independent and when there is no replacement the event is considered dependent. Homework Questions: page 311 #16 - 20 VENN DIAGRAMS - a way to graphically represent sets and set events Each diagram begins with a rectangle representing the Sample Space S (often called the universal set) and then each event is represented by a circle. Complement of Event A ( A ) Event A U A U A B U A B Event A or B U Event B B Event A and B U Applying Venn Diagrams to Probability: There are 4 people that own a cat AND a dog – Bill, Jane, Bob and Lori. Example: 20 people have been asked if they owned a cat, a dog, or both. Cat Bill Jane Bob Lori Helen Lynn Tom Hugh Chris Bill Jane Bob Lori Dog Erin Liam Ashley Chelsey Sara Amanda Zach Luke Cat U 5 Dog 4 Note: 5 4 8 3 20 8 3 There are 9 people that own a cat. How many people own a cat or a dog or both? There are 3 people that own neither a cat nor a dog. How many people own just a dog? How many people do not own a cat? If you can answer these questions, then you can determine the probabilities of these events happening. Example: P(Cat ) 9 or 0.45 20 Text Book: Page 312 – 315 P ( Not a Cat ) 11 or 0.55 20 These events are complements and together total a probability of one! Start with Investigation 5 and work through until #36 page 315 Important Formulas: P( A or B) P( A) P( B) P( A and B) P( A or B) P( A) P( B) P( A and B) P( A) P( B) Do you know what the difference is?