The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore

Bedford Freeman Worth Publishers

CHAPTER 5Probability: What Are the Chances?5.1

Randomness, Probability, and Simulation

Learning Objectives

After this section, you should be able to:

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INTERPRET probability as a long-run relative frequency.

USE simulation to MODEL chance behavior.

Randomness, Probability, and Simulation

What is randomness?Pick a number:

1 2 3 4What did you pick?Almost 75% of people will pick 3. 20% pick 2 or 4. Only about 5% choose 1!

Give an example of a false positive:

Give an example of a false negative:

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The Idea of Probability

Chance behavior is unpredictable in the _________, but has a regular and ____________ in the long run.The law of large numbers says that if we observe more and more ________ of any chance process, the proportion of times that a specific outcome occurs approaches a single value.

The probability of any outcome of a chance process is a number between _________ that describes the proportion of times the outcome would occur in a very _____ series of repetitions.

Suppose that 4 friends get together to study at Tim’s house for their next test in AP Statistics. When they go for a snack in the kitchen, Tim’s three-year-old brother makes a tower using their textbooks. Unfortunately, none of the students wrote his name in the book, so when they leave each student takes one of the books at random. When the students returned the books at the end of the year and the clerk scanned their barcodes, the students were surprised that none of the four had their own book. How likely is it that none of the four students ended up with the correct book?

http://www.rossmanchance.com/applets/randomBabies/Babies.html

Another way to interpret probability of an outcome is its predicted long-run relative frequency. For example, if we do many trials of flipping a fair coin, we would expect to see the proportion of heads to be about .5.

BUT each trial is completely random and not based on any previous flip or set of flips.

Horse race simulation: We are using the sum of the numbers on a roll of 2 die to simulate horses moving around a track. You can choose to be horse # 2, 3, 4, ..., 12.

Which number would you choose? Why?

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Myths About Randomness

The idea of probability seems straightforward. However, there are some myths of chance behavior we must address.

The myth of short-run regularity:

The idea of probability is that randomness is predictable in the long run . Our intuition tries to tell us random phenomena should also be predictable in the short run. However, probability does not allow us to make short-run predictions.

The myth of the “ law of averages ” :

Probability tells us random behavior evens out in the long run. Future outcomes are not affected by past behavior. That is, past outcomes do not influence the likelihood of individual outcomes occurring in the future.

What are some myths about randomness?Myth: Random events are predictable in the short run.

Myth: A "hot hand" indicates that a streak is likely to continue.

Myth: The "Law of Averages" says a streak makes other outcomes more likely.

Truth: Random events ARE predictable in the long run. Truth: Coins, dice, cards, etc. have no memories. LLN is long run.Truth

Myth

Myth

Myth

Imagine you are flipping a coin. Write down the results of 50 imaginary flips (e.g. HTTHT…):

Use technology to simulate: (Write down the steps and results here)

What is the longest run in each set?

HW page 300 (1, 3, 8, 9, 11, 37, 38)

Dear Abby,My husband and I just had our 8th child. Another girl, and I am really one disappointed woman. I suppose i should thank God she was healthy, but Abby, this one was supposed to have been a boy. Even the doctor told me that the law of averages was in our favor 100 to one."Abigail Van Buren, 1974

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Simulation

The __________ of chance behavior, based on a model that accurately reflects the situation, is called a simulation .

State : Ask a question of interest about some chance process.

Plan : Describe how to use a chance device to imitate one repetition of the process. Tell what you will record at the end of each repetition.

Do : Perform many repetitions of the simulation.

Conclude : Use the results of your simulation to answer the question of interest.

Performing a Simulation

We can use physical devices, random numbers (e.g. Table D), and technology to perform simulations.

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Example: Simulations with technology

In an attempt to increase sales, a breakfast cereal company decides to offer a NASCAR promotion. Each box of cereal will contain a collectible card featuring one of these NASCAR drivers: Jeff Gordon, Dale Earnhardt, Jr., Tony Stewart, Danica Patrick, or Jimmie Johnson.

The company says that each of the 5 cards is equally likely to appear in any box of cereal.

A NASCAR fan decides to keep buying boxes of the cereal until she has all 5 drivers’cards. She is surprised when it takes her 23 boxes to get the full set of cards. Should she be surprised?

Problem : What is the probability that it will take 23 or more boxes to get a full set of 5 NASCAR collectible cards?

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

The Practice of Statistics, 5 th

Example: Simulations with technology

Plan : We need five numbers to represent the five possible cards. Let’s let 1 = Jeff Gordon, 2 = Dale Earnhardt, Jr., 3 = Tony Stewart, 4 = Danica Patrick, and 5 = Jimmie Johnson.

We’ll use randInt(1,5) to simulate buying one box of cereal and looking at which card is inside.

Because we want a full set of cards, we’ll keep pressing Enter until we get all five of the labels from 1 to 5. We’ll record the number of boxes that we had to open.

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Example: Simulations with technology

Conclude : We never had to buy more than 22 boxes to get the full set of NASCAR drivers’cards in 50 repetitions of our simulation. So our estimate of the probability that it takes 23 or more boxes to get a full set is roughly 0. The NASCAR fan should be surprised about how many boxes she had to buy.

Suppose I want to choose a simple random sample of size 6 from a group of 60 seniors and 30 juniors. To do this, I write each person’s name on an equally sized piece of paper and mix them up in a large grocery bag. Just as I am about to select the first name, a thoughtful student suggests that I should stratify by class. I agree, and we decide it would be appropriate to select 4 seniors and 2 juniors. However, since I already mixed up the names, I don’t want to have separate them all again. Can I just draw names until I get 4 seniors and 2 juniors?

Design and carry out a simulation using Table D to estimate the probability that you must draw 8 or more names to get 4 seniors and 2 juniors.

What are some common errors when using a table of random numbers?

Answ

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Section Summary

In this section, we learned how to…

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$ INTERPRET probability as a long-run relative frequency.

$ USE simulation to MODEL chance behavior.

Randomness, Probability, and Simulation