Chapter 5 (NEW)

Chapter 5 - Probability TEACHER: Mr. Gottschalk SUBJECT: AP Statistics DAY CLASSWORK HOMEWORK 0 Chapter 4 Test W&TN Video #1 [Simulations] 1 Quiz 5.1 Worksheet (pg. 5 – 6) W&TN Video #2 [Probability Models] 2 p. 297 (#31 – 36) + p. 309 (#43, 46) W&TN Video #3 – part 1 [General Addition Rule] 3 p. 309 (#49b, 50b, 51, 57, 58, 60) W&TN Video #3 – part 2 [General Addition Rule] 4 Quiz 5.2 Worksheet (pg. 11 – 13) W&TN Video #4 [Conditional Prob. + Indep. Events] 5 p. 329 (#64, 66, 67, 69, 72, 106) W&TN Video #5 – part 1 [Gen. Multiplication Rule] 6 p. 330 (#83, 85, 92, 99) W&TN Video #5 – part 2 [Gen. Multiplication Rule] 7 Quiz 5.3 Worksheet (pg. 18 – 20) FRAPPY Day Tomorrow!!! 8 FRAPPY DAY Work on Semester Project or Practice WS A/B/C on pg. 21 – 23 with answers on pg. 24 9 Probability Review – Day #1 (MC) Look @ p. 334-335 in your book for extra problems Watch BONUS MULTIPLE CHOICE REVIEW VIDEO!!! 10 Probability Review – Day #2 (FR) Look @ p. 334-335 in your book for extra problems Watch BONUS MULTIPLE CHOICE REVIEW VIDEO!!! 11 Chapter 5 Test W&TN Video #1 of Ch.6 (Discrete Random Variables) W&TN = W atch & T ake N otes Lesson Objectives: Throughout this chapter we will focus on learning the rules of probability and how to use them to calculate the probability of various types of events. We will also learn the terminology of probability, such as, independent and dependent events, mutually exclusive (or disjoint ) events, union , intersection , and joint probabilities . Probability calculations can be difficult for many students, so we will do many practice problems to gain more experience and in turn a better understanding of how to approach probability problems. 1

Chapter 5 Topics Page Number 5.1 - Simulations ...................................................................................................... 3 5.2 – Probability Models .......................................................................................... 7 5.2 – The General Addition Rule, Venn Diagrams, Two-Way Tables, etc ................ 9 5.3 – Conditional Probability and Independence ................................................... 14 5.3 – The General Multiplication Rule and Tree Diagrams .................................... 16 2

I used random.org’s random number generator to come up with 100 sets of three numbers, either 0 or 1. _______________ times, the simulation showed 3 girls. The _______________ the series of repetitions, the closer we get to the actual probability of the event. So, the probability of having three girls out of three children is ____________________ to 11%! This idea is referred to as the LAW OF _____________________ _____________________________. Ex : A basketball player has made 82% of his shots this season. What is the probability that he makes 10 out of 10 shots in the next game? Assumptions: each shot is independepent & P(make one shot) = 0.82. The plan to carry out this simulation:  Randomly generate __________ numbers between 1-100 to represent the ten free-throw attempts.  Let _________________ = made shot and __________________ = missed shot .  Repeat _____________ times and calculate the proportion of times that all 10 shots are made. In ten separate trials of 1000 games each, the probability of making all 10 shots is: 0.139, 0.121, 0.152, 0.131, 0.145, 0.145, 0.156, 0.142, 0.131, and 0.145. Conclusion : If our 82% free-throw shooter attempts 10 free-throws ___________, ___________ __________, the probability that he makes all 10 shots is close to ____________. ************************************************************************************ ****** Imagine 50% of people like vanilla, 30% like chocolate, and 20% like both ice cream flavors. Describe a simulation plan to choose TWO people’s favorite ice cream flavor. (Don’t worry, you won’t actually carry out the simulation, just plan it!) 4

Quiz 5.1 AP Statistics 1. The probability that a randomly selected person in the United States is left- handed is about 0.14. (a)Interpret this probability. (b) Among the 28 students in Mr. Millar’s Calculus BC class, 8 are left- handed. Could this have happened by chance alone? Describe how you would use a random number table to simulate the proportion of left-handers in a class of 28 students if they were chosen randomly from a population that is 14% left-handed. Do not perform the simulation. Below are the number of left-handers in 100 simulated classes of 28 students each, assuming that students are selected randomly from a population in which 14% of individuals are left-handed. 5 Remember, this is how many actual lefties are in Mr. Millar’s class!

Chapter 5: [Video #2] – Probability Models To help us organize events and their probabilities, we can create a probability model . Probability model = __________________________________________________________ __________________________________________________________ A probability model is made up of two parts:  Sample space : _________________________________________________________ _________________________________________________________  Probability per outcome The sample space can have categorical or quantitative variables:  Coin flip: { _____________ , ______________ }  Rolling a die: { _____ , _____ , _____ , _____ , _____ , _____ }  Suit of playing card: { ____________ , ____________ , ___________ , ____________ }  Sum of two dice: { ___ , ___ , ___ , ___ , ___ , ____ , ____ , ____ , ____ , ____ , ____ } 7

Probability models are commonly set up with a table with the sample space of events in one row and the associated probabilities beneath each event. Ex : Rolling a single six-sided die. Ex : Benford’s Law states that in many naturally occurring collections of numbers, the leading significant digit is likely to be small.  What if we did not know the probability of the first number being a 9? 8 Probability Model RULES!!! 1) 2)

 Event #2: Spade ( S ) o P( S ) = _______  Event #3: Jack or Spade ( J or S ) o P( J or S ) = _______ ******************************************************************************************************************* Does the same pattern from before still hold true? _______ What makes these three events different than before??? _____________________________________________________________ ***General Addition Rule*** _______________________________________________________________________  If two (or more) events have nothing in common, then these events are called _____________ or __________________ _________________________.  Therefore, if events “A” and “B” are disjoint, then P(A and B) = ________ ******************************************************************************************************************* Using two-way tables of probabilities to answer the same questions as before… … using disjoint events… or …using non-disjoint events. ******************************************************************************************************************* Chapter 5: [Video #3 – continued!] – Probability Rules (Part 1 of 3) Alternate way of writing the GENERAL ADDITION RULE using logic symbols… 10

P( __________ ) = P(A) + P(B) – P( ____________ ) Ex : Elsa and Anna auditioned to be a part of the school play. No one auditioned for a specific spot yet. Elsa believes she has a 0.4 chance of making it. Anna believes she has a 0.6 chance of making it. They believe they both have a 0.3 chance of making it. Make a Venn diagram that shows all the events and probabilities.  ********************************************************* 1) What is the probability that only Anna is selected to be in the play? 2) What is the probability that at least one of the two girls are selected? Quiz 5.2 AP Statistics 1. The table below is a probability model for the number of cars in a randomly- selected household in the United States. (Based on U.S. Census 2000 data). (a) What is the probability that a randomly selected household has three cars? (That is, fill in the space marked with a “?”) Show your work. (b) What is the probability that a randomly-selected household has at least 2 cars? Show your work. 11

(a)What is the probability that a person selected randomly earned a Professional or Doctoral degree? (b) What type of events are “Professional” and “Doctoral”? How do you know? (c) What is the probability that a person selected randomly is female or earned a Master’s degree? (d) What type of events are “female” and “Master’s”? How do you know? 4. Consolidated Builders has bid on two large construction contracts. The company president believes that the probability of winning the first contract (event A ) is 0.6, that the probability of winning a second (event B ) is 0.3, and that the probability of winning both jobs is 0.1. (a) Construct BOTH a Venn diagram AND a two-way table that summarizes events A and B. A A C B 13

B C (b) Write each of the following events in terms of A , B , A C , and B C , and use the information above to calculate the probability of each. i. Consolidated wins both jobs. ii. Consolidated wins the first job but not the second. iii. Consolidated wins at least one of the jobs. iv . Either Consolidated does not win the first job or wins the second job. Chapter 5: [Video #4] – Probability Rules (Part 2 of 3) I collected data from my students about if they had a part-time job and whether they pay for their own car insurance. o P( Yes pays own ins. ) = o P( Yes pays own ins. | Yes Job) = … wait a second … 14

 We can say that having a job and paying his/her own insurance are _______________________________ events all because mathematically… ___________________________________________________________________. **************************************************************************************************  If your parent was a student athlete, does that affect whether you would be a student athlete, too??? o P( Yes student ) = o P( Yes student | Yes parent ) = o P( Yes student | No parent ) =  Ans: _______, if the parent was an athlete or not ______ ______ affect the probability the student is an athlete.  We can conclude your parent’s decision to be a student athlete and your decision to be a student athlete are _________________________________________ events all because mathematically … ________________________________________________________________________________________. ************************************************************************************************** In conclusion…  Conditional probability formula:  Check for independent events: ************************************************************************************************** 1) What is the probability that Anna is selected to be in the play, given Elsa is selected to be in the play? 2) Are the events “Anna is selected” and “Elsa is selected” independent from one another? Chapter 5: [Video #5] – Probability Rules (Part 3 of 3) We have been using Venn diagrams and two-way tables to organize our data to answer probability questions, but those work great when discussing a single event with multiple possibilities. Now, we move on to more complex probability questions that involve multiple events with multiple possibilities ? ****************************************************************************************** In Texas Hold’em poker, two cards are dealt to each player. The best starting hand is two aces. Draw a tree diagram showing all the possibilities of getting aces vs. no aces for each card dealt. 16

General Multiplication Rule o What’s the probability that you are dealt two aces in a hand? o What’s the probability that you are dealt one ace in a hand? o What’s the probability that you are dealt at least one ace in a hand? Chapter 5: [Video #5 – continued!] – Probability Rules (Part 3 of 3) Ex #2 : When a football team gets the ball within the 20 yard line, the coach has three play options: throw a long pass, throw a short pass, or run the ball. The play will result in a touchdown or not. P( long pass ) = 0.15 P( long pass and no touchdown ) = 0.1275 P( touchdown | short pass ) = 0.12 P( run and touchdown ) = 0.02 P( no touchdown | run ) = 0.96 17

Suppose one subject from this study was selected at random. (a) Calculate the probability that the selected subject preferred Twitter. (b) Calculate the probability that the selected subject preferred Facebook & was in the 25-44 age group. (c) Calculate the probability that the selected subject preferred LinkedIn or was in the 0-24 age group. (d) Calculate the probability that the selected subject preferred Twitter, given that he or she was in the 45 – 64 age group. (e) Are the events “preferred Twitter” and “age group 45 – 64” independent? Provide mathematical evidence. (f) Are the events “preferred Twitter” and “age group 45 – 64” mutually exclusive? Explain. 19

2. Some days, Ramon drives to work. The rest of the time he rides his bike. Suppose we choose a random work day. The following table gives the probabilities of several events. (a) Draw a complete tree diagram with all events and their probabilities. (b) Calculate the probability that Ramon is not late for work given that he bikes. (c) Calculate the probability that Ramon is late for work given that he drives. (d) Calculate the probability that Ramon drives given that he is late. (e) Calculate the probability that Ramon bikes and he is not late. 20