Waiting Times. Lesson1. Unit UNIT 7 PATTERNS IN CHANCE

Size: px
Start display at page:

Download "Waiting Times. Lesson1. Unit UNIT 7 PATTERNS IN CHANCE"

Transcription

1 Lesson1 Waiting Times Monopoly is a board game that can be played by several players. Movement around the board is determined by rolling a pair of dice. Winning is based on a combination of chance and a sense for making smart real estate deals. While playing Monopoly, Anita draws the card shown below. She must go directly to the jail space on the board. Anita may get out of jail by rolling doubles with a pair of dice on one of her next three turns. Doubles means that both dice show the same number on the top. If she does not roll doubles on any of the three turns, Anita must pay a $50 fine to get out of jail. Anita takes her first turn and doesn t roll doubles. On her second turn, she doesn t roll doubles again. On her third and final try, Anita doesn t roll doubles yet again. She grudgingly pays the $50 to get out of jail. Anita is feeling very unlucky. 45 UNIT 7 PATTERNS IN CHANCE 45 UNIT 7 PATTERNS IN CHANCE

2 Lesson1 Waiting Times LESSON OVERVIEW One of the most common probabilistic situations is waiting for a specific event to occur: waiting for red to come up in roulette, waiting for doubles to appear in backgammon or Monopoly, waiting for a day of rain on days when the weather forecaster says there is a 20 percent chance of rain, etc. In this lesson, students will be introduced to the idea of a waiting-time distribution (also called the geometric distribution). They will construct frequency distributions and histograms of waiting-time distributions and discover that all have the same basic shape. In a standard waiting-time situation, the probability of a success on each trial is assumed to be the same, no matter what happened on previous trials. Students will explore situations in which this assumption is true and in which it is not. L Obj t To use simulation to construct frequency distributions for waiting-time situations when trials are independent, and to recognize the shape as skewed to the right To identify when trials are independent To recognize rare events To review finding the mean of a frequency distribution To review finding the probabilities of events associated with rolling a pair of dice LAUNCH full-class discussion You might want to begin this lesson by bringing in a Monopoly game and explaining the rules about how to get out of jail by rolling doubles. Present the situation on page 45 and let your students discuss it, using the Think About This Situation questions to facilitate the discussion. This will give you an opportunity to assess how much your students know about probability. LESSON 1 WAITING TIMES T45

3 Master Think About This Situation Transparency Master LAUNCH continued Think About This Situation i Use with page 457. Anita s situation suggests several questions. How likely is it that a Monopoly player who is sent to jail (and doesn t have a Get Out of Jail Free card) will have to pay $50 to leave? As a class, think of as many ways to find the answer to this question as you can. In games and in real life, people are occasionally in the position of waiting for an event to happen. In some cases, the event becomes more and more likely to happen with each opportunity. In some cases, the event becomes less and less likely to happen with each opportunity. How does the chance of rolling doubles change each time Anita rolls the dice? On average, how many rolls do you think it takes to roll doubles? Do you think Anita should feel unlucky? Explain your reasoning. Master Steps in a Simulation U N I T 7 P A TTERNS IN CHANCE 1. Be sure you understand the problem state it in your own words. 2. Identify all possible outcomes and determine the probability of each. 3. State the assumptions you are making. 4. Select a random device and describe how it models your problem. 5. Conduct one trial, recording the result in a frequency table.. Do a large number of trials, recording the results in your frequency table and a histogram. 7. Summarize your results and give your conclusion. Use with page 457. Master Name Rolling Doubles Rolled doubles on first try 1 Rolled doubles on second try 2 Rolled doubles on third try 3 Rolled doubles on fourth try 4 Transparency Master U N I T 7 P A TTERNS IN CHANCE Date Activity Master Event Number of Rolls Frequency Total 100 See Teaching Master 149. a Have students record their conjectures to the how likely question so they can compare this with their result to Activity 1, Part d. You might also make a list of the ways students suggest to get the answer to this question. Some students surely will suggest trying the experiment of rolling dice. At this stage, students aren t expected to know how to do this problem theoretically, and many may not know that the probability of rolling doubles on one roll of a pair of dice is 1. b Although your students may know that Anita s chance of rolling doubles remains the same each time she rolls the dice (because that is what they have been told), many of them won t really believe it. If you are interested in the psychology of learning probability and want to test this, take a coin and try the following experiment with your students. If you don t discuss the correct answer at this time, you can come back to this scenario while discussing the final unit checkpoint. Students should be able to answer the questions correctly at that time. Ask, If I flip a coin ten times, how many heads do I expect to get, on the average? (Most students will answer five. Be sure that they understand that, on the average, half of the flips of a fair coin will be heads.) Flip the coin and announce that you got a head. Flip it again and announce that you got another head. This shouldn t seem strange to the class. Now ask them, When I have finished the ten flips, how many heads do you expect me to have? Almost every student will answer five. This answer indicates that the students expect the coin to balance out the first two flips of heads. In other words, they believe that tails are due and now have a probability greater than 1. There are eight 2 flips left and they expect only three of them to be heads. The correct answer is six. You have eight flips left and, with a fair coin, you expect four of them to be heads and four to be tails. With the first two flips of heads, that means you expect six heads for the ten flips. c Students may suggest trying an experiment or base their hunches on their game-playing experiences. They won t know that the theoretical answer is, but you don t need to tell them yet. (Students might want to look back at this question and see how the results from Activity 1 might help them decide whether or not Anita should feel unlucky.) Use with page 457. U N I T 7 P A TTERNS IN CHANCE EXPLORE INVESTIGATION 1 small-group investigation Waiting for Doubles See Teaching Master 150. By the end of this investigation, students will have constructed their first probability distribution, used it to give some estimates of probabilities, and calculated some theoretical answers for related questions. As you circulate among the groups, encourage them to describe what they are doing and to talk about the differences between finding experimental (Activity 2) and theoretical (Activity 3) probabilities. You also might remind them of the initial question, On average, how many rolls does it take to get doubles? See additional Teaching Notes on page T533C. T457 UNIT 7 PATTERNS IN CHANCE

4 Thi k Ab Thi Si i Anita s situation suggests several questions. a b c How likely is it that a Monopoly player who is sent to jail (and doesn t have a Get Out of Jail Free card) will have to pay $50 to leave? As a class, think of as many ways to find the answer to this question as you can. In games and in real life, people are occasionally in the position of waiting for an event to happen. In some cases, the event becomes more and more likely to happen with each opportunity. In some cases, the event becomes less and less likely to happen with each opportunity. Does the chance of rolling doubles change each time Anita rolls the dice? On average, how many rolls do you think it takes to roll doubles? Do you think Anita should feel unlucky? Explain your reasoning. INVESTIGATION 1 Waiting for Doubles In this investigation, you will explore several aspects of Anita s situation. For this investigation, we will change the rules of Monopoly so that a player must stay in jail until he or she rolls doubles. A player cannot pay $50 to get out of jail in this version of the game, and there is no Get Out of Jail Free card. 1. Now suppose you are playing Monopoly under this new rule and have just been sent to jail. Take your first turn and roll a pair of dice. Did you roll doubles and get out of jail? If so, stop. If not, roll again. Did you roll doubles and get out of jail on your second turn? If so, stop. If not, roll again. Did you roll doubles and get out of jail on your third turn? If so, stop. If not, keep rolling until you get doubles. a. Copy the frequency table below and put a tally mark in the frequency column next to the event that happened to you. Add rows as needed. Rolling Doubles Event Number of Rolls Frequency Rolled doubles on first try 1 Rolled doubles on second try 2 Rolled doubles on third try 3 Rolled doubles on fourth try 4 Total 100 LESSON 1 WAITING TIMES 457 LESSON 1 WAITING TIMES 457

5 b. With other members of your class, perform this experiment a total of 100 times. Record the results in your frequency table. c. Do the events in the frequency table appear to be equally likely? That is, does each of the events have the same chance of happening? d. Use your frequency table to estimate the probability that Anita will have to pay $50, or use a Get Out of Jail Free card, to get out of jail when playing a standard version of Monopoly. Compare this estimate with your original estimate in Part a of the Think About This Situation on page 457. e. Make a histogram of the data in your frequency table. Describe the shape of this histogram. f. Explain why the frequencies in your table are decreasing even though the probability of rolling doubles on each attempt does not change. 2. Later in this unit, you will analyze Anita s situation theoretically. That is, you will use mathematical principles to find the probability she has to pay $50. As a first step, in this activity you will explore how to find the probability of various events when two dice are rolled. a. Suppose a red die and a green die are rolled at the same time. Make a copy of the matrix-like chart below. R lli g T Number on Red Die Di Number on Green Die , 1 3, 2 4, 5 What does the entry 3, 2 mean? Complete the chart, showing all possible outcomes when the two dice are rolled. How many outcomes are possible? Are these outcomes equally likely? Why or why not? 458 UNIT 7 PATTERNS IN CHANCE 458 UNIT 7 PATTERNS IN CHANCE

6 EXPLORE continued 1. c. The events in the table are not equally likely. The probability of rolling doubles for the first time on that roll decreases with each roll. This is reflected in the table through the decreasing frequencies. d. Probabilities will vary based on the results of the experiment. Using the sample frequency table from Part b, students will find that the estimated probability that Anita will get out of jail in the first three rolls is or Thus, the estimated 100 probability she will have to pay $50 is or (The theoretical probability is 0.58, as students will learn later.) Comparisons will depend on the students original estimate. e. Histograms will vary. A sample histogram follows; most histograms won t be this smooth, and they usually will continue on to the right. Rolling Doubles Masters 152a 152b Name 152a Rolling Two Dice Number on Red Die Use with page Date Number on Green Die , 1 3, 2 4, 5 Activity Master UNIT 7 PATTERNS IN CHANCE 20 Frequency Number of Trials The histogram is skewed right. The bars decrease in height. Each bar is about 5 of the height of the bar to its left. f. There will be fewer people who will roll doubles for the first time on their second roll than who will roll doubles for the first time on their first roll. One way to picture this is to imagine all 100 people taking their first roll simultaneously. Those who get doubles on this roll (about 1 of the 100 people) leave the room. Everyone who remains then tries a second time to get doubles. About 1 will succeed. This isn t 1 of 100, but 1 of a smaller number. 2. See Teaching Masters 152a 152b. a. The 3, 2 entry means that when the dice are rolled, there is a 3 on the red die and a 2 on the green die. See additional Teaching Notes on page T533C. LESSON 1 WAITING TIMES T458

7 Master Use with page 459. Checkpoint Transparency Master In this investigation, you explored the waiting time for rolling doubles. Suppose you compared your class s histogram of the waiting time for rolling doubles with another class s histogram. Explain why the histograms should or should not be exactly the same. What characteristics do you think the histograms will have in common? If two dice are rolled several times, what is the probability of getting doubles on the first roll? On the fourth roll? Be prepared to share your ideas with the entire class. U N I T 7 P A TTERNS IN CHANCE EXPLORE continued 2. b. Students should use the table they just constructed to compute these figures It may be helpful if you point out that it is not always the best policy to write fractions in the lowest terms when computing probabilities. For example, it s much easier to compare the probability of rolling a sum of 7 with that of rolling a sum of 11 if both probabilities have a denominator of 3. c. Yes, the probabilities are the same. The color simply helps organize the table of all possible outcomes. d. Each has a 1 chance of rolling doubles on her next turn. SHARE AND SUMMARIZE full-class discussion If your students are not sure of the response to Part b of the Checkpoint on page 459, you may wish to have them carry out the following experiment. The workload should be divided among groups of students. One group of students will represent Conchita trying to roll doubles on her first roll of the dice. That group will roll the dice many times, for example, 400 times, each time a first try, and count the number of doubles. This group now has an approximation of the probability of rolling doubles on the first try. The second group, representing Anita, would roll the dice until they did not get doubles twice in a row. The group would then try to get doubles on the third roll. Counting only the times the group tried a third roll, they should repeat this process 400 times (not doubles, not doubles, try to roll doubles). This group now has an approximation of the probability of rolling doubles on the third try, given that doubles didn t happen on the first or second try. The two groups probably won t get exactly the same probability. Before the experiment, discuss with the class how close the relative frequencies will have to be so that they believe the theoretical probabilities are the same and how far apart the relative frequencies will have to be so that they believe the theoretical probabilities are different. (With 400 repetitions, there is a 95 percent chance that the probabilities will be within 0.05 of each other.) One of the difficulties of simulation is the large number of trials needed to convince students that two probabilities are the same. If, for example, one student flips a nickel 100 times and another flips a penny 100 times, there is a very good chance that there will be a difference of 0.10 or more in the proportion of heads with the nickel and with the penny. One of the students could easily get 45 heads out of 100 and the other get 5 out of 100, for a difference of Very roughly, if each experiment is repeated n times, there is a 95 percent chance that the two estimates will be no farther apart than 2. n See additional Teaching Notes on page T533D. T459 UNIT 7 PATTERNS IN CHANCE

8 b. If two dice are rolled, what is the probability of getting each of the following events? Doubles A sum of 7 A sum of 11 A sum of 7 or a sum of 11 Either a 2 on one or both dice or a sum of 2 c. Is the probability of rolling doubles the same if both dice are the same color? Explain your reasoning. d. Suppose that in playing the modified Monopoly game, Anita is still in jail after trying twice to roll doubles. Conchita has just been sent to jail. Does Anita or Conchita have a better chance of rolling doubles on her next turn? Compare your answer with that of other groups. Resolve any differences. Ch kp i In this investigation, you explored the waiting time for rolling doubles. a b Suppose you compared your class s histogram of the waiting time for rolling doubles with another class s histogram. Explain why the histograms should or should not be exactly the same. What characteristics do you think the histograms will have in common? If two dice are rolled several times, what is the probability of getting doubles on the first roll? On the fourth roll? Be prepared to share your ideas with the entire class. O Yo O Change the rules of Monopoly so that a player must flip a coin and get heads in order to get out of jail. a. Is it harder or easier to get out of jail with this new rule instead of by rolling doubles? Explain your reasoning. b. Play this version 24 times, either with a coin or by simulating the situation. Put your results in a table like the one in Activity 1 of Investigation 1. Then make a histogram of your results. c. What is your estimate of the probability that a player will get out of jail in three flips or fewer? LESSON 1 WAITING TIMES 459 LESSON 1 WAITING TIMES 459

Junior Circle Meeting 5 Probability. May 2, ii. In an actual experiment, can one get a different number of heads when flipping a coin 100 times?

Junior Circle Meeting 5 Probability. May 2, ii. In an actual experiment, can one get a different number of heads when flipping a coin 100 times? Junior Circle Meeting 5 Probability May 2, 2010 1. We have a standard coin with one side that we call heads (H) and one side that we call tails (T). a. Let s say that we flip this coin 100 times. i. How

More information

Key Concepts. Theoretical Probability. Terminology. Lesson 11-1

Key Concepts. Theoretical Probability. Terminology. Lesson 11-1 Key Concepts Theoretical Probability Lesson - Objective Teach students the terminology used in probability theory, and how to make calculations pertaining to experiments where all outcomes are equally

More information

Multiplication and Probability

Multiplication and Probability Problem Solving: Multiplication and Probability Problem Solving: Multiplication and Probability What is an efficient way to figure out probability? In the last lesson, we used a table to show the probability

More information

Name Class Date. Introducing Probability Distributions

Name Class Date. Introducing Probability Distributions Name Class Date Binomial Distributions Extension: Distributions Essential question: What is a probability distribution and how is it displayed? 8-6 CC.9 2.S.MD.5(+) ENGAGE Introducing Distributions Video

More information

Part 1: I can express probability as a fraction, decimal, and percent

Part 1: I can express probability as a fraction, decimal, and percent Name: Pattern: Part 1: I can express probability as a fraction, decimal, and percent For #1 to #4, state the probability of each outcome. Write each answer as a) a fraction b) a decimal c) a percent Example:

More information

Date. Probability. Chapter

Date. Probability. Chapter Date Probability Contests, lotteries, and games offer the chance to win just about anything. You can win a cup of coffee. Even better, you can win cars, houses, vacations, or millions of dollars. Games

More information

10-4 Theoretical Probability

10-4 Theoretical Probability Problem of the Day A spinner is divided into 4 different colored sections. It is designed so that the probability of spinning red is twice the probability of spinning green, the probability of spinning

More information

Such a description is the basis for a probability model. Here is the basic vocabulary we use.

Such a description is the basis for a probability model. Here is the basic vocabulary we use. 5.2.1 Probability Models When we toss a coin, we can t know the outcome in advance. What do we know? We are willing to say that the outcome will be either heads or tails. We believe that each of these

More information

Unit 6: What Do You Expect? Investigation 2: Experimental and Theoretical Probability

Unit 6: What Do You Expect? Investigation 2: Experimental and Theoretical Probability Unit 6: What Do You Expect? Investigation 2: Experimental and Theoretical Probability Lesson Practice Problems Lesson 1: Predicting to Win (Finding Theoretical Probabilities) 1-3 Lesson 2: Choosing Marbles

More information

Probability A = {(1,4), (2,3), (3,2), (4,1)},

Probability A = {(1,4), (2,3), (3,2), (4,1)}, Probability PHYS 1301 F99 Prof. T.E. Coan version: 15 Sep 98 The naked hulk alongside came, And the twain were casting dice; The game is done! I ve won! I ve won! Quoth she, and whistles thrice. Samuel

More information

Lesson Lesson 3.7 ~ Theoretical Probability

Lesson Lesson 3.7 ~ Theoretical Probability Theoretical Probability Lesson.7 EXPLORE! sum of two number cubes Step : Copy and complete the chart below. It shows the possible outcomes of one number cube across the top, and a second down the left

More information

Simulations. 1 The Concept

Simulations. 1 The Concept Simulations In this lab you ll learn how to create simulations to provide approximate answers to probability questions. We ll make use of a particular kind of structure, called a box model, that can be

More information

Data Analysis and Numerical Occurrence

Data Analysis and Numerical Occurrence Data Analysis and Numerical Occurrence Directions This game is for two players. Each player receives twelve counters to be placed on the game board. The arrangement of the counters is completely up to

More information

1. For which of the following sets does the mean equal the median?

1. For which of the following sets does the mean equal the median? 1. For which of the following sets does the mean equal the median? I. {1, 2, 3, 4, 5} II. {3, 9, 6, 15, 12} III. {13, 7, 1, 11, 9, 19} A. I only B. I and II C. I and III D. I, II, and III E. None of the

More information

WORKSHOP SIX. Probability. Chance and Predictions. Math Awareness Workshops

WORKSHOP SIX. Probability. Chance and Predictions. Math Awareness Workshops WORKSHOP SIX 1 Chance and Predictions Math Awareness Workshops 5-8 71 Outcomes To use ratios and a variety of vocabulary to describe the likelihood of an event. To use samples to make predictions. To provide

More information

Lesson 8: The Difference Between Theoretical Probabilities and Estimated Probabilities

Lesson 8: The Difference Between Theoretical Probabilities and Estimated Probabilities Lesson 8: The Difference Between Theoretical Probabilities and Estimated Probabilities Did you ever watch the beginning of a Super Bowl game? After the traditional handshakes, a coin is tossed to determine

More information

A Probability Work Sheet

A Probability Work Sheet A Probability Work Sheet October 19, 2006 Introduction: Rolling a Die Suppose Geoff is given a fair six-sided die, which he rolls. What are the chances he rolls a six? In order to solve this problem, we

More information

Grade 8 Math Assignment: Probability

Grade 8 Math Assignment: Probability Grade 8 Math Assignment: Probability Part 1: Rock, Paper, Scissors - The Study of Chance Purpose An introduction of the basic information on probability and statistics Materials: Two sets of hands Paper

More information

Probability MAT230. Fall Discrete Mathematics. MAT230 (Discrete Math) Probability Fall / 37

Probability MAT230. Fall Discrete Mathematics. MAT230 (Discrete Math) Probability Fall / 37 Probability MAT230 Discrete Mathematics Fall 2018 MAT230 (Discrete Math) Probability Fall 2018 1 / 37 Outline 1 Discrete Probability 2 Sum and Product Rules for Probability 3 Expected Value MAT230 (Discrete

More information

Math 147 Lecture Notes: Lecture 21

Math 147 Lecture Notes: Lecture 21 Math 147 Lecture Notes: Lecture 21 Walter Carlip March, 2018 The Probability of an Event is greater or less, according to the number of Chances by which it may happen, compared with the whole number of

More information

Section 6.1 #16. Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

Section 6.1 #16. Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? Section 6.1 #16 What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? page 1 Section 6.1 #38 Two events E 1 and E 2 are called independent if p(e 1

More information

Basic Probability Ideas. Experiment - a situation involving chance or probability that leads to results called outcomes.

Basic Probability Ideas. Experiment - a situation involving chance or probability that leads to results called outcomes. Basic Probability Ideas Experiment - a situation involving chance or probability that leads to results called outcomes. Random Experiment the process of observing the outcome of a chance event Simulation

More information

Math 1313 Section 6.2 Definition of Probability

Math 1313 Section 6.2 Definition of Probability Math 1313 Section 6.2 Definition of Probability Probability is a measure of the likelihood that an event occurs. For example, if there is a 20% chance of rain tomorrow, that means that the probability

More information

If a fair coin is tossed 10 times, what will we see? 24.61% 20.51% 20.51% 11.72% 11.72% 4.39% 4.39% 0.98% 0.98% 0.098% 0.098%

If a fair coin is tossed 10 times, what will we see? 24.61% 20.51% 20.51% 11.72% 11.72% 4.39% 4.39% 0.98% 0.98% 0.098% 0.098% Coin tosses If a fair coin is tossed 10 times, what will we see? 30% 25% 24.61% 20% 15% 10% Probability 20.51% 20.51% 11.72% 11.72% 5% 4.39% 4.39% 0.98% 0.98% 0.098% 0.098% 0 1 2 3 4 5 6 7 8 9 10 Number

More information

Probability. Ms. Weinstein Probability & Statistics

Probability. Ms. Weinstein Probability & Statistics Probability Ms. Weinstein Probability & Statistics Definitions Sample Space The sample space, S, of a random phenomenon is the set of all possible outcomes. Event An event is a set of outcomes of a random

More information

TEKSING TOWARD STAAR MATHEMATICS GRADE 7. Hands-on-Activity. Six Weeks 3

TEKSING TOWARD STAAR MATHEMATICS GRADE 7. Hands-on-Activity. Six Weeks 3 TEKSING TOWARD STAAR MATHEMATICS GRADE 7 Hands-on-Activity Six Weeks 3 TEKSING TOWARD STAAR 2014 Six Weeks 3 Lesson 4 Teacher Notes for Student Activity 3 MATERIALS: Per Pair of Students: 1 bag of 4 colored

More information

Bellwork Write each fraction as a percent Evaluate P P C C 6

Bellwork Write each fraction as a percent Evaluate P P C C 6 Bellwork 2-19-15 Write each fraction as a percent. 1. 2. 3. 4. Evaluate. 5. 6 P 3 6. 5 P 2 7. 7 C 4 8. 8 C 6 1 Objectives Find the theoretical probability of an event. Find the experimental probability

More information

PRE TEST KEY. Math in a Cultural Context*

PRE TEST KEY. Math in a Cultural Context* PRE TEST KEY Salmon Fishing: Investigations into A 6 th grade module in the Math in a Cultural Context* UNIVERSITY OF ALASKA FAIRBANKS Student Name: PRE TEST KEY Grade: Teacher: School: Location of School:

More information

* How many total outcomes are there if you are rolling two dice? (this is assuming that the dice are different, i.e. 1, 6 isn t the same as a 6, 1)

* How many total outcomes are there if you are rolling two dice? (this is assuming that the dice are different, i.e. 1, 6 isn t the same as a 6, 1) Compound probability and predictions Objective: Student will learn counting techniques * Go over HW -Review counting tree -All possible outcomes is called a sample space Go through Problem on P. 12, #2

More information

PRE TEST. Math in a Cultural Context*

PRE TEST. Math in a Cultural Context* P grade PRE TEST Salmon Fishing: Investigations into A 6P th module in the Math in a Cultural Context* UNIVERSITY OF ALASKA FAIRBANKS Student Name: Grade: Teacher: School: Location of School: Date: *This

More information

What Do You Expect? Concepts

What Do You Expect? Concepts Important Concepts What Do You Expect? Concepts Examples Probability A number from 0 to 1 that describes the likelihood that an event will occur. Theoretical Probability A probability obtained by analyzing

More information

Lesson 8: The Difference Between Theoretical Probabilities and Estimated Probabilities

Lesson 8: The Difference Between Theoretical Probabilities and Estimated Probabilities Lesson 8: The Difference Between Theoretical and Estimated Student Outcomes Given theoretical probabilities based on a chance experiment, students describe what they expect to see when they observe many

More information

The Teachers Circle Mar. 20, 2012 HOW TO GAMBLE IF YOU MUST (I ll bet you $5 that if you give me $10, I ll give you $20.)

The Teachers Circle Mar. 20, 2012 HOW TO GAMBLE IF YOU MUST (I ll bet you $5 that if you give me $10, I ll give you $20.) The Teachers Circle Mar. 2, 22 HOW TO GAMBLE IF YOU MUST (I ll bet you $ that if you give me $, I ll give you $2.) Instructor: Paul Zeitz (zeitzp@usfca.edu) Basic Laws and Definitions of Probability If

More information

Data Analysis and Probability

Data Analysis and Probability Data Analysis and Probability Vocabulary List Mean- the sum of a group of numbers divided by the number of addends Median- the middle value in a group of numbers arranged in order Mode- the number or item

More information

Lesson 4: Calculating Probabilities for Chance Experiments with Equally Likely Outcomes

Lesson 4: Calculating Probabilities for Chance Experiments with Equally Likely Outcomes NYS COMMON CORE MAEMAICS CURRICULUM 7 : Calculating Probabilities for Chance Experiments with Equally Likely Classwork Examples: heoretical Probability In a previous lesson, you saw that to find an estimate

More information

Probability, Continued

Probability, Continued Probability, Continued 12 February 2014 Probability II 12 February 2014 1/21 Last time we conducted several probability experiments. We ll do one more before starting to look at how to compute theoretical

More information

CSC/MTH 231 Discrete Structures II Spring, Homework 5

CSC/MTH 231 Discrete Structures II Spring, Homework 5 CSC/MTH 231 Discrete Structures II Spring, 2010 Homework 5 Name 1. A six sided die D (with sides numbered 1, 2, 3, 4, 5, 6) is thrown once. a. What is the probability that a 3 is thrown? b. What is the

More information

a) Getting 10 +/- 2 head in 20 tosses is the same probability as getting +/- heads in 320 tosses

a) Getting 10 +/- 2 head in 20 tosses is the same probability as getting +/- heads in 320 tosses Question 1 pertains to tossing a fair coin (8 pts.) Fill in the blanks with the correct numbers to make the 2 scenarios equally likely: a) Getting 10 +/- 2 head in 20 tosses is the same probability as

More information

Lesson 4: Calculating Probabilities for Chance Experiments with Equally Likely Outcomes

Lesson 4: Calculating Probabilities for Chance Experiments with Equally Likely Outcomes Lesson : Calculating Probabilities for Chance Experiments with Equally Likely Outcomes Classwork Example : heoretical Probability In a previous lesson, you saw that to find an estimate of the probability

More information

Author(s): Hope Phillips

Author(s): Hope Phillips Title: Game Show Math Real-World Connection: Grade: 8th Author(s): Hope Phillips BIG Idea: Probability On the game show, The Price is Right, Plinko is a favorite! According to www.thepriceisright.com the

More information

CS1802 Week 9: Probability, Expectation, Entropy

CS1802 Week 9: Probability, Expectation, Entropy CS02 Discrete Structures Recitation Fall 207 October 30 - November 3, 207 CS02 Week 9: Probability, Expectation, Entropy Simple Probabilities i. What is the probability that if a die is rolled five times,

More information

EECS 203 Spring 2016 Lecture 15 Page 1 of 6

EECS 203 Spring 2016 Lecture 15 Page 1 of 6 EECS 203 Spring 2016 Lecture 15 Page 1 of 6 Counting We ve been working on counting for the last two lectures. We re going to continue on counting and probability for about 1.5 more lectures (including

More information

The study of probability is concerned with the likelihood of events occurring. Many situations can be analyzed using a simplified model of probability

The study of probability is concerned with the likelihood of events occurring. Many situations can be analyzed using a simplified model of probability The study of probability is concerned with the likelihood of events occurring Like combinatorics, the origins of probability theory can be traced back to the study of gambling games Still a popular branch

More information

Fraction Race. Skills: Fractions to sixths (proper fractions) [Can be adapted for improper fractions]

Fraction Race. Skills: Fractions to sixths (proper fractions) [Can be adapted for improper fractions] Skills: Fractions to sixths (proper fractions) [Can be adapted for improper fractions] Materials: Dice (2 different colored dice, if possible) *It is important to provide students with fractional manipulatives

More information

Mathematics Behind Game Shows The Best Way to Play

Mathematics Behind Game Shows The Best Way to Play Mathematics Behind Game Shows The Best Way to Play John A. Rock May 3rd, 2008 Central California Mathematics Project Saturday Professional Development Workshops How much was this laptop worth when it was

More information

Tail. Tail. Head. Tail. Head. Head. Tree diagrams (foundation) 2 nd throw. 1 st throw. P (tail and tail) = P (head and tail) or a tail.

Tail. Tail. Head. Tail. Head. Head. Tree diagrams (foundation) 2 nd throw. 1 st throw. P (tail and tail) = P (head and tail) or a tail. When you flip a coin, you might either get a head or a tail. The probability of getting a tail is one chance out of the two possible outcomes. So P (tail) = Complete the tree diagram showing the coin being

More information

I. WHAT IS PROBABILITY?

I. WHAT IS PROBABILITY? C HAPTER 3 PROAILITY Random Experiments I. WHAT IS PROAILITY? The weatherman on 10 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and

More information

November 11, Chapter 8: Probability: The Mathematics of Chance

November 11, Chapter 8: Probability: The Mathematics of Chance Chapter 8: Probability: The Mathematics of Chance November 11, 2013 Last Time Probability Models and Rules Discrete Probability Models Equally Likely Outcomes Probability Rules Probability Rules Rule 1.

More information

What is the expected number of rolls to get a Yahtzee?

What is the expected number of rolls to get a Yahtzee? Honors Precalculus The Yahtzee Problem Name Bolognese Period A Yahtzee is rolling 5 of the same kind with 5 dice. The five dice are put into a cup and poured out all at once. Matching dice are kept out

More information

!"#$%&'("&)*("*+,)-(#'.*/$'-0%$1$"&-!!!"#$%&'(!"!!"#$%"&&'()*+*!

!#$%&'(&)*(*+,)-(#'.*/$'-0%$1$&-!!!#$%&'(!!!#$%&&'()*+*! !"#$%&'("&)*("*+,)-(#'.*/$'-0%$1$"&-!!!"#$%&'(!"!!"#$%"&&'()*+*! In this Module, we will consider dice. Although people have been gambling with dice and related apparatus since at least 3500 BCE, amazingly

More information

Review. Natural Numbers: Whole Numbers: Integers: Rational Numbers: Outline Sec Comparing Rational Numbers

Review. Natural Numbers: Whole Numbers: Integers: Rational Numbers: Outline Sec Comparing Rational Numbers FOUNDATIONS Outline Sec. 3-1 Gallo Name: Date: Review Natural Numbers: Whole Numbers: Integers: Rational Numbers: Comparing Rational Numbers Fractions: A way of representing a division of a whole into

More information

Activity 8. Gambler s Fallacy: Lucky Streaks and Slumps. Objective. Introduction. Problem

Activity 8. Gambler s Fallacy: Lucky Streaks and Slumps. Objective. Introduction. Problem Objective Activity 8 Determine the probability of independent, compound events Design simulations and collect data to explore streaking behavior Introduction The expression, I m on a roll! is often heard

More information

Random Variables. A Random Variable is a rule that assigns a number to each outcome of an experiment.

Random Variables. A Random Variable is a rule that assigns a number to each outcome of an experiment. Random Variables When we perform an experiment, we are often interested in recording various pieces of numerical data for each trial. For example, when a patient visits the doctor s office, their height,

More information

Problem Set 9: The Big Wheel... OF FISH!

Problem Set 9: The Big Wheel... OF FISH! Opener David, Jonathan, Mariah, and Nate each spin the Wheel of Fish twice.. The Wheel is marked with the numbers 1, 2, 3, and 10. Players earn the total number of combined fish from their two spins. 1.

More information

Exam III Review Problems

Exam III Review Problems c Kathryn Bollinger and Benjamin Aurispa, November 10, 2011 1 Exam III Review Problems Fall 2011 Note: Not every topic is covered in this review. Please also take a look at the previous Week-in-Reviews

More information

1. Theoretical probability is what should happen (based on math), while probability is what actually happens.

1. Theoretical probability is what should happen (based on math), while probability is what actually happens. Name: Date: / / QUIZ DAY! Fill-in-the-Blanks: 1. Theoretical probability is what should happen (based on math), while probability is what actually happens. 2. As the number of trials increase, the experimental

More information

Chapter 8: Probability: The Mathematics of Chance

Chapter 8: Probability: The Mathematics of Chance Chapter 8: Probability: The Mathematics of Chance Free-Response 1. A spinner with regions numbered 1 to 4 is spun and a coin is tossed. Both the number spun and whether the coin lands heads or tails is

More information

Georgia Department of Education Georgia Standards of Excellence Framework GSE Geometry Unit 6

Georgia Department of Education Georgia Standards of Excellence Framework GSE Geometry Unit 6 How Odd? Standards Addressed in this Task MGSE9-12.S.CP.1 Describe categories of events as subsets of a sample space using unions, intersections, or complements of other events (or, and, not). MGSE9-12.S.CP.7

More information

MATH STUDENT BOOK. 7th Grade Unit 6

MATH STUDENT BOOK. 7th Grade Unit 6 MATH STUDENT BOOK 7th Grade Unit 6 Unit 6 Probability and Graphing Math 706 Probability and Graphing Introduction 3 1. Probability 5 Theoretical Probability 5 Experimental Probability 13 Sample Space 20

More information

Random Variables. Outcome X (1, 1) 2 (2, 1) 3 (3, 1) 4 (4, 1) 5. (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) }

Random Variables. Outcome X (1, 1) 2 (2, 1) 3 (3, 1) 4 (4, 1) 5. (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) } Random Variables When we perform an experiment, we are often interested in recording various pieces of numerical data for each trial. For example, when a patient visits the doctor s office, their height,

More information

Use the following games to help students practice the following [and many other] grade-level appropriate math skills.

Use the following games to help students practice the following [and many other] grade-level appropriate math skills. ON Target! Math Games with Impact Students will: Practice grade-level appropriate math skills. Develop mathematical reasoning. Move flexibly between concrete and abstract representations of mathematical

More information

Discrete Structures for Computer Science

Discrete Structures for Computer Science Discrete Structures for Computer Science William Garrison bill@cs.pitt.edu 6311 Sennott Square Lecture #23: Discrete Probability Based on materials developed by Dr. Adam Lee The study of probability is

More information

This Probability Packet Belongs to:

This Probability Packet Belongs to: This Probability Packet Belongs to: 1 2 Station #1: M & M s 1. What is the sample space of your bag of M&M s? 2. Find the theoretical probability of the M&M s in your bag. Then, place the candy back into

More information

2 Event is equally likely to occur or not occur. When all outcomes are equally likely, the theoretical probability that an event A will occur is:

2 Event is equally likely to occur or not occur. When all outcomes are equally likely, the theoretical probability that an event A will occur is: 10.3 TEKS a.1, a.4 Define and Use Probability Before You determined the number of ways an event could occur. Now You will find the likelihood that an event will occur. Why? So you can find real-life geometric

More information

Foundations to Algebra In Class: Investigating Probability

Foundations to Algebra In Class: Investigating Probability Foundations to Algebra In Class: Investigating Probability Name Date How can I use probability to make predictions? Have you ever tried to predict which football team will win a big game? If so, you probably

More information

Ex 1: A coin is flipped. Heads, you win $1. Tails, you lose $1. What is the expected value of this game?

Ex 1: A coin is flipped. Heads, you win $1. Tails, you lose $1. What is the expected value of this game? AFM Unit 7 Day 5 Notes Expected Value and Fairness Name Date Expected Value: the weighted average of possible values of a random variable, with weights given by their respective theoretical probabilities.

More information

Probability. March 06, J. Boulton MDM 4U1. P(A) = n(a) n(s) Introductory Probability

Probability. March 06, J. Boulton MDM 4U1. P(A) = n(a) n(s) Introductory Probability Most people think they understand odds and probability. Do you? Decision 1: Pick a card Decision 2: Switch or don't Outcomes: Make a tree diagram Do you think you understand probability? Probability Write

More information

The Coin Toss Experiment

The Coin Toss Experiment Experiments p. 1/1 The Coin Toss Experiment Perhaps the simplest probability experiment is the coin toss experiment. Experiments p. 1/1 The Coin Toss Experiment Perhaps the simplest probability experiment

More information

Stat 20: Intro to Probability and Statistics

Stat 20: Intro to Probability and Statistics Stat 20: Intro to Probability and Statistics Lecture 17: Using the Normal Curve with Box Models Tessa L. Childers-Day UC Berkeley 23 July 2014 By the end of this lecture... You will be able to: Draw and

More information

A referee flipped a fair coin to decide which football team would start the game with

A referee flipped a fair coin to decide which football team would start the game with Probability Lesson.1 A referee flipped a fair coin to decide which football team would start the game with the ball. The coin was just as likely to land heads as tails. Which way do you think the coin

More information

Most of the time we deal with theoretical probability. Experimental probability uses actual data that has been collected.

Most of the time we deal with theoretical probability. Experimental probability uses actual data that has been collected. AFM Unit 7 Day 3 Notes Theoretical vs. Experimental Probability Name Date Definitions: Experiment: process that gives a definite result Outcomes: results Sample space: set of all possible outcomes Event:

More information

Grade 6 Math Circles Fall Oct 14/15 Probability

Grade 6 Math Circles Fall Oct 14/15 Probability 1 Faculty of Mathematics Waterloo, Ontario Centre for Education in Mathematics and Computing Grade 6 Math Circles Fall 2014 - Oct 14/15 Probability Probability is the likelihood of an event occurring.

More information

Find the probability of an event by using the definition of probability

Find the probability of an event by using the definition of probability LESSON 10-1 Probability Lesson Objectives Find the probability of an event by using the definition of probability Vocabulary experiment (p. 522) trial (p. 522) outcome (p. 522) sample space (p. 522) event

More information

Probability is often written as a simplified fraction, but it can also be written as a decimal or percent.

Probability is often written as a simplified fraction, but it can also be written as a decimal or percent. CHAPTER 1: PROBABILITY 1. Introduction to Probability L EARNING TARGET: I CAN DETERMINE THE PROBABILITY OF AN EVENT. What s the probability of flipping heads on a coin? Theoretically, it is 1/2 1 way to

More information

What are the chances?

What are the chances? What are the chances? Student Worksheet 7 8 9 10 11 12 TI-Nspire Investigation Student 90 min Introduction In probability, we often look at likelihood of events that are influenced by chance. Consider

More information

COMPOUND EVENTS. Judo Math Inc.

COMPOUND EVENTS. Judo Math Inc. COMPOUND EVENTS Judo Math Inc. 7 th grade Statistics Discipline: Black Belt Training Order of Mastery: Compound Events 1. What are compound events? 2. Using organized Lists (7SP8) 3. Using tables (7SP8)

More information

Basic Probability Concepts

Basic Probability Concepts 6.1 Basic Probability Concepts How likely is rain tomorrow? What are the chances that you will pass your driving test on the first attempt? What are the odds that the flight will be on time when you go

More information

Lenarz Math 102 Practice Exam # 3 Name: 1. A 10-sided die is rolled 100 times with the following results:

Lenarz Math 102 Practice Exam # 3 Name: 1. A 10-sided die is rolled 100 times with the following results: Lenarz Math 102 Practice Exam # 3 Name: 1. A 10-sided die is rolled 100 times with the following results: Outcome Frequency 1 8 2 8 3 12 4 7 5 15 8 7 8 8 13 9 9 10 12 (a) What is the experimental probability

More information

Raise your hand if you rode a bus within the past month. Record the number of raised hands.

Raise your hand if you rode a bus within the past month. Record the number of raised hands. 166 CHAPTER 3 PROBABILITY TOPICS Raise your hand if you rode a bus within the past month. Record the number of raised hands. Raise your hand if you answered "yes" to BOTH of the first two questions. Record

More information

Making Middle School Math Come Alive with Games and Activities

Making Middle School Math Come Alive with Games and Activities Making Middle School Math Come Alive with Games and Activities For more information about the materials you find in this packet, contact: Sharon Rendon (605) 431-0216 sharonrendon@cpm.org 1 2-51. SPECIAL

More information

A. 15 B. 24 C. 45 D. 54

A. 15 B. 24 C. 45 D. 54 A spinner is divided into 8 equal sections. Lara spins the spinner 120 times. It lands on purple 30 times. How many more times does Lara need to spin the spinner and have it land on purple for the relative

More information

PATTERNS IN CHANCE. Many events in life. Lessons. 1 Calculating Probabilities. 2 Modeling Chance Situations

PATTERNS IN CHANCE. Many events in life. Lessons. 1 Calculating Probabilities. 2 Modeling Chance Situations UNIT 8 Many events in life occur by chance. They cannot be predicted with certainty. For example, you cannot predict whether the next baby born will be a boy or a girl. However, in the long run, you can

More information

MEI Conference Short Open-Ended Investigations for KS3

MEI Conference Short Open-Ended Investigations for KS3 MEI Conference 2012 Short Open-Ended Investigations for KS3 Kevin Lord Kevin.lord@mei.org.uk 10 Ideas for Short Investigations These are some of the investigations that I have used many times with a variety

More information

Probability Rules. 2) The probability, P, of any event ranges from which of the following?

Probability Rules. 2) The probability, P, of any event ranges from which of the following? Name: WORKSHEET : Date: Answer the following questions. 1) Probability of event E occurring is... P(E) = Number of ways to get E/Total number of outcomes possible in S, the sample space....if. 2) The probability,

More information

Data Collection Sheet

Data Collection Sheet Data Collection Sheet Name: Date: 1 Step Race Car Game Play 5 games where player 1 moves on roles of 1, 2, and 3 and player 2 moves on roles of 4, 5, # of times Player1 wins: 3. What is the theoretical

More information

A C E. Answers Investigation 3. Applications. 12, or or 1 4 c. Choose Spinner B, because the probability for hot dogs on Spinner A is

A C E. Answers Investigation 3. Applications. 12, or or 1 4 c. Choose Spinner B, because the probability for hot dogs on Spinner A is Answers Investigation Applications. a. Answers will vary, but should be about for red, for blue, and for yellow. b. Possible answer: I divided the large red section in half, and then I could see that the

More information

Determine the Expected value for each die: Red, Blue and Green. Based on your calculations from Question 1, do you think the game is fair?

Determine the Expected value for each die: Red, Blue and Green. Based on your calculations from Question 1, do you think the game is fair? Answers 7 8 9 10 11 12 TI-Nspire Investigation Student 120 min Introduction Sometimes things just don t live up to their expectations. In this activity you will explore three special dice and determine

More information

Essential Question How can you list the possible outcomes in the sample space of an experiment?

Essential Question How can you list the possible outcomes in the sample space of an experiment? . TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G..B Sample Spaces and Probability Essential Question How can you list the possible outcomes in the sample space of an experiment? The sample space of an experiment

More information

b) Find the exact probability of seeing both heads and tails in three tosses of a fair coin. (Theoretical Probability)

b) Find the exact probability of seeing both heads and tails in three tosses of a fair coin. (Theoretical Probability) Math 1351 Activity 2(Chapter 11)(Due by EOC Mar. 26) Group # 1. A fair coin is tossed three times, and we would like to know the probability of getting both a heads and tails to occur. Here are the results

More information

Normal Distribution Lecture Notes Continued

Normal Distribution Lecture Notes Continued Normal Distribution Lecture Notes Continued 1. Two Outcome Situations Situation: Two outcomes (for against; heads tails; yes no) p = percent in favor q = percent opposed Written as decimals p + q = 1 Why?

More information

Conditional Probability Worksheet

Conditional Probability Worksheet Conditional Probability Worksheet P( A and B) P(A B) = P( B) Exercises 3-6, compute the conditional probabilities P( AB) and P( B A ) 3. P A = 0.7, P B = 0.4, P A B = 0.25 4. P A = 0.45, P B = 0.8, P A

More information

Counting Methods and Probability

Counting Methods and Probability CHAPTER Counting Methods and Probability Many good basketball players can make 90% of their free throws. However, the likelihood of a player making several free throws in a row will be less than 90%. You

More information

Section Theoretical and Experimental Probability...Wks 3

Section Theoretical and Experimental Probability...Wks 3 Name: Class: Date: Section 6.8......Theoretical and Experimental Probability...Wks 3. Eight balls numbered from to 8 are placed in a basket. One ball is selected at random. Find the probability that it

More information

CS1800: Intro to Probability. Professor Kevin Gold

CS1800: Intro to Probability. Professor Kevin Gold CS1800: Intro to Probability Professor Kevin Gold Probability Deals Rationally With an Uncertain World Using probabilities is the only rational way to deal with uncertainty De Finetti: If you disagree,

More information

Outcome X (1, 1) 2 (2, 1) 3 (3, 1) 4 (4, 1) 5 {(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)}

Outcome X (1, 1) 2 (2, 1) 3 (3, 1) 4 (4, 1) 5 {(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)} Section 8: Random Variables and probability distributions of discrete random variables In the previous sections we saw that when we have numerical data, we can calculate descriptive statistics such as

More information

Section 1: Data (Major Concept Review)

Section 1: Data (Major Concept Review) Section 1: Data (Major Concept Review) Individuals = the objects described by a set of data variable = characteristic of an individual weight height age IQ hair color eye color major social security #

More information

CSI 23 LECTURE NOTES (Ojakian) Topics 5 and 6: Probability Theory

CSI 23 LECTURE NOTES (Ojakian) Topics 5 and 6: Probability Theory CSI 23 LECTURE NOTES (Ojakian) Topics 5 and 6: Probability Theory 1. Probability Theory OUTLINE (References: 5.1, 5.2, 6.1, 6.2, 6.3) 2. Compound Events (using Complement, And, Or) 3. Conditional Probability

More information

Name: Class: Date: 6. An event occurs, on average, every 6 out of 17 times during a simulation. The experimental probability of this event is 11

Name: Class: Date: 6. An event occurs, on average, every 6 out of 17 times during a simulation. The experimental probability of this event is 11 Class: Date: Sample Mastery # Multiple Choice Identify the choice that best completes the statement or answers the question.. One repetition of an experiment is known as a(n) random variable expected value

More information

Probability. Sometimes we know that an event cannot happen, for example, we cannot fly to the sun. We say the event is impossible

Probability. Sometimes we know that an event cannot happen, for example, we cannot fly to the sun. We say the event is impossible Probability Sometimes we know that an event cannot happen, for example, we cannot fly to the sun. We say the event is impossible Impossible In summer, it doesn t rain much in Cape Town, so on a chosen

More information

Probability. The Bag Model

Probability. The Bag Model Probability The Bag Model Imagine a bag (or box) containing balls of various kinds having various colors for example. Assume that a certain fraction p of these balls are of type A. This means N = total

More information