Ch. 11 Simulations.notebook

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November 14, 2019

Aug 13­1:38 PM

Warm-upIf the standard deviation of a set of observations is 0, you can conclude:

A) that there is no relationship between the observations.

B) that the average value is 0.C) that all the observations are the same value.D) that a mistake in arithmetic has been made.E) none of the above.

CAnswer:

Sep 24­9:18 PM

UNIT 4: Probability, Random Variables, and Probability Distributions

Ch. 11: Simulations

Ch. 11 Simulations.notebook

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November 14, 2019

Sep 20­9:55 AM

Why Be Random?

• What is it about random outcomes being random that makes random selection seem fair?> Nobody can guess the outcome before it happens.> When we want things to be fair, usually some

underlying set of outcomes will be equally likely (although in many games some combinations of outcomes are more likely than others).

• Example:> Pick a number 1­4

Humans are not good at being random!

Sep 20­9:57 AM

It's Not Easy Being Random

• Computers have become a popular way to generate random numbers.> Since computers follow programs, the "random"

numbers we get from the computers are really pseudorandom.

> Pseudorandom values are good enough for most purposes.

• We also use random number tables.• Keep in mind that even though we may see a pattern, patterns do not necessarily mean that variation is not random.

Ch. 11 Simulations.notebook

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November 14, 2019

Sep 20­10:01 AM

TI Tips­ Generating Random Numbers

• MATH• PRB• 5: randInt(first integer, last integer, number of integers)

Nov 5­10:01 AM

A random process generates results that are determined by chance.

An outcome is the result of a trial of a random process.

ex: Rolling a particular value on a six­sided number cube is one of six possible outcomes.

An event is a collection of outcomes.

ex: When rolling two six­sided number cubes, an event would be a sum of seven. The corresponding collection of outcomes would be (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1), where the ordered pairs indicate (face value on one cube, face value on the other cube).

Ch. 11 Simulations.notebook

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November 14, 2019

Oct 22­4:21 PM

Simulationa simulation is a way to model random events, such that

simulated outcomes closely match real­world outcomes.

We use random­digit outcomes to mimic the uncertainty of a response variable of interest.

**We use simulations to help us investigate a question for which many outcomes are possible, we can't or don't want to collect data and a mathematical answer is hard to calculate.

Nov 5­10:14 AM

In a simulation, all possible outcomes are associated with a value to be determined by chance. We then record the counts of simulated outcomes and count the total.

To estimate the probability of an outcome or an event happening, we use the relative frequency from the simulation or empirical data.

The law of large numbers states that simulated (empirical) probabilities tend to get closer to the true probability as the number of trials increase.

ex: roll a die 10 times, then 20 times, then 50 times and record your data.

Let's do this in our calculator too!

https://apcentral.collegeboard.org/courses/ap­statistics/classroom­resources/graphing­calculator­simulations­simplified?course=ap­statistics

Ch. 11 Simulations.notebook

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November 14, 2019

Nov 7­9:22 AM

Sep 20­10:02 AM

A Simulation

• We are going to use random numbers to simulate reality.• The sequence of events we want to investigate is called a trial.

• The basic building block of a simulation is called a component.> Trials usually involve several components

• After the trial, we record what happened ­ our response variable.

• There are seven steps to a simulation...

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November 14, 2019

Sep 20­10:06 AM

Simulation Steps

1. Identify the component to be repeated.2. Explain how you will model the component's outcome.3. Explain how you will combine the components to model

a trial.4. State clearly what the response variable is.5. Run several trials.6. Collect and summarize the results of all the trials.7. State your conclusion.

Nov 13­8:42 AM

Example: Suppose a cereal manufacturer puts pictures of famous athletes on cards in boxes of cereal in hope to boost sales. The manufacturer announces the 20% contain a picture of Tiger Woods, 30% of David Beckham, and the rest contain Serena Williams. You want all three. How many boxes do you expect to buy before you get a complete set?

1. The component is opening a box of cereal.

2. We will use the numbers 0­9 to represent the outcome of opening the box. Because 20% of the cards are Tiger Woods, we'll let 0­1 represent Tiger. Because 30% of the cards have David Beckham, we'll let 2­4 represent Beckham, and since the rest are Serena Williams, we'll let the rest of the digits, 5­9 represent Serena.

3. We will keep "opening boxes" (repeating components) until our collection is complete. We do this by looking at each random digit and indicating what picture it represents. We will continue until we have found all three.

4. The response variable is the number of boxes it will take to complete the set.

5. Run Several Trials.

6. Collect and summarize the results of all the trials.

7. Based on this simulation, we estimate that customers hoping to complete their card collections will need to open a median of boxes, although it could take many more.

Ch. 11 Simulations.notebook

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November 14, 2019

Nov 13­8:42 AM

Example: Suppose a cereal manufacturer puts pictures of famous athletes on cards in boxes of cereal in hope to boost sales. The manufacturer announces the 20% contain a picture of Tiger Woods, 30% of David Beckham, and the rest contain Serena Williams. You want all three. How many boxes do you expect to buy before you get a complete set?

1. The component is opening a box of cereal.

2. We will use the numbers 0­9 to represent the outcome of opening the box. Because 20% of the cards are Tiger Woods, we'll let 0­1 represent Tiger. Because 30% of the cards have David Beckham, we'll let 2­4 represent Beckham, and since the rest are Serena Williams, we'll let the rest of the digits, 5­9 represent Serena.

3. We will keep "opening boxes" (repeating components) until our collection is complete. We do this by looking at each random digit and indicating what picture it represents. We will continue until we have found all three.

4. The response variable is the number of boxes it will take to complete the set.

5. Run Several Trials.

6. Collect and summarize the results of all the trials.

7. Based on this simulation, we estimate that customers hoping to complete their card collections will need to open a median of boxes, although it could take many more.

Oct 22­11:32 AM

What Can Go Wrong:­ Remember, a simulation is just a model. We didn't really buy any cereal ­ so beware of confusing what really happens with the conclusion that a simulation suggests might happen. Real results will not match simulate results exactly.

­ Make sure to model your situation correctly. Look carefully at the digits that you're looking for to make sure that they match your situation. If your simulation overlooks important aspects of the real situation, your model will not be accurate.

­ Run enough trials!! ­ Don't be fooled by the results of only 5 or 10 trials. With the help of computer software or calculators, we can do many trials. For now, stick with a large number of trials.

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November 14, 2019

Sep 20­10:50 AM

Warm­upIn a certain community, 20% of cable subscribers also subscribe to the company's broadband service for their Internet connection. You would like to design a simulation to estimate the probability that one of the six randomly selected subscribers has the broadband service. Using digits 0 through 9, which of the following assignments would be appropriate to model this situation?

(A) Assign even digits to broadband subscribers & odd digits to cable­only subscribers.(B) Assign 0 and 1 to broadband subscribers and 2­9 to cable­only subscribers.(C) Assign 0, 1, and 2 to broadband subscribers and 3­9 to cable­only subscribers.(D) Assign 1­6 to broadband subscribers and 7, 8, 9, and 0 to cable­only subscribers.(E) Assign 0, 1, and 2 to broadband subscribers; 3, 4, 5, and 6 to cable­only subscribers; and ignore digits 7, 8, and 9.

B

Nov 13­8:23 AM

Example: On the average, how many girls would you expect in a family of three children?7 1 0 3 5 0 9 0 0 1 4 3 3 6 7 4 9 4 9 7 7 2 7 1 9 9 6 7 5 8

2 7 6 1 1 9 1 5 9 6 5 4 5 8 0 8 1 5 0 7 2 7 1 0 2 5 6 0 2 7

1. The component is the birth of a child

2. I will look at one digit random numbers. Let 0­4 represent the birth of a girl, and 5­9 represent the birth of a boy.

3. Each trial consists of identifying 3 digits as G (girl) or B (boy).

4. The response variable is the number of girls in a family of three.

5. Let's run several trials.

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November 14, 2019

Sep 20­10:10 AM

Example: You take a quiz with 5 multiple choice questions. After you studied, you estimated that you would have about a 75% chance of getting any individual question right. What are your chances of getting them all right? Use at least 20 trials.

Plan: State the problem. Identify the important parts of your simulation.

Components: Identify the components.Outcomes: State how you will model each component uing equally likely random digits.

Trial: Explain how you will combine the components to simulate a trial.

Response Variable: Define your response variable.Mechanics: Run at least 20 trials.Analyze: Summarize the results across all trials to answer the initial question.

Sep 20­10:10 AM

Example: You take a quiz with 5 multiple choice questions. After you studied, you estimated that you would have about a 75% chance of getting any individual question right. What are your chances of getting them all right? Use at least 20 trials.

Plan: I'll use a simulation to investigate whether it's unlikely that you would get all 5 questions correct on a multiple choice test.

Components: A component is answering one multiple choice question.

Outcomes: I will look at one­digit random numbers between 0­3 using my calculator.

Let 0­2 represent getting a question correct. Let 3 represent getting the question wrong.

Trial: Each trial consists of identifying digits 0­3 as C (correct answer) or W (wrong answer) until 5 digits are chosen (5 questions answered). Then I will determine if all questions were correct and mark it in my chart.

Response Variable: The response variable is whether or not you answer all 5 multiple choice questions correctly.

Mechanics: Run at least 20 trials.Trial # Component Outcomes All Correct?

Analyze: Summarize the results across all trials to answer the initial question.

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November 14, 2019

Sep 20­10:14 AM

Example: 57 students participated in a lottery for a particularly desirable dorm room ­ a triple with a fireplace and private bath in the tower. 20 of the participants were member of the same varsity team. When all three winners were members of the team, the other students cried foul.

Could an all­team outcome reasonably be expected to happen if everyone had a fair shot at the room.

Sep 20­10:14 AM

Example: 57 students participated in a lottery for a particularly desirable dorm room ­ a triple with a fireplace and private bath in the tower. 20 of the participants were member of the same varsity team. When all three winners were members of the team, the other students cried foul.

Could an all­team outcome reasonably be expected to happen if everyone had a fair shot at the room.

Ch. 11 Simulations.notebook

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November 14, 2019

Sep 23­9:20 AM

Warm UpThe average yearly snowfall in a city is 55 inches. What is the standard deviation if 15% of the years have snowfalls above 60 inches? Assume yearly snowfalls are normally distributed.(A) 4.83(B) 5.18(C) 6.04(D) 8.93(E) The standard deviation cannot be computed from the information given.

A

Sep 20­10:24 AM

Close

The baseball World Series consists of up to seven games. The first team to win four games wins the series. The first two are played at one team's home ballpark, the next three at the other team's park, and the final two (if needed) are played back at the first park. Records over the past century show that there is a home field advantage; the home team has about a 55% chance of winning. Does the current system of alternating ballparks even out the home field advantage? How often will the team that begins at home win the series?

1. What is the component to be repeated?

2. How will you model each component from equally likely random digits?

3. How will you model a trial by combining components?

4. What is the response variable?

5. How will you analyze the response variable?

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November 14, 2019

Sep 23­9:19 AM

Agenda:

1) Simulations Practice

2) If you finish early­ begin reading Ch. 14

Sep 18­11:15 PM