Chapter 5: Probability: What are the Chances? Section 5.1 Randomness, Probability, and Simulation The Practice of Statistics, 4th edition – For AP* STARNES, YATES, MOORE

Chapter 5 Probability: What Are the Chances? 5.1 Randomness, Probability, and Simulation 5.2 Probability Rules 5.3 Conditional Probability and Independence

Section 5.1 Randomness, Probability, and Simulation Learning Objectives After this section, you should be able to… DESCRIBE the idea of probability DESCRIBE myths about randomness DESIGN and PERFORM simulations

Randomness, Probability, and Simulation Whose Book is This? 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? On four equally sized slips of paper, write the numbers 1, 2, 3, 4. Shuffle the papers and lay them down one at a time in a row. If the number on the paper matches it’s position in the row (e.g. paper 2 ends up in the second position), this represents a student choosing his own book from the tower of textbooks. Count the number of students who get the correct book. Repeat this several more times, recording the number of students who get the correct book in each trial. Combine your results with your classmates and estimate how often none of the four end up with their own book. Randomness, Probability, and Simulation

Randomness, Probability, and Simulation The Idea of Probability Chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long run. The law of large numbers says that if we observe more and more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches a single value. Randomness, Probability, and Simulation Definition: The probability of any outcome of a chance process is a number between 0 (never occurs) and 1(always occurs) that describes the proportion of times the outcome would occur in a very long series of repetitions.

Randomness, Probability, and Simulation Whose Book is This? The graphs below show the short-run and long-run behavior of the proportion of trials in which there are no matches when 4 students choose a book at random. The blue line is the correct probability of 0.375. As you can see, in the first 20 trials, there is quite a bit of variability. However, after 500 trials, the proportion of times there was no match is quite close to the actual value. Randomness, Probability, and Simulation

Randomness, Probability, and Simulation Extended Warranties How much should a company charge for an extended warranty for a specific type of cell phone? Suppose that 5% of these cell phones under warranty will be returned and the cost to replace the phone is $150. If the company knew which phones would go bad, it could charge $150 for these phones and $0 for the rest. However, since the company can’t know which phones will be returned but knows that about 1 in every 20 will be returned, they should charge at least 150/20 = $7.50 for the extended warranty. Randomness, Probability, and Simulation

Randomness, Probability, and Simulation Myths about Randomness The idea of probability seems straightforward. However, there are several myths of chance behavior we must address. Randomness, Probability, and Simulation 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.

Randomness, Probability, and Simulation Ex: Runs in Die Rolling Ex: Red is Due! Randomness, Probability, and Simulation Roll a die 12 times and record the result of each roll. Which of the following outcomes is more probable? 123456654321 154524336126 These outcomes are both equally (un)likely, even though the first set of rolls has a more noticeable pattern. In casinos, there is often a large display next to every roulette table showing the outcomes of the last several spins of the wheel. Since the results of previous spins reveal nothing about the results of future spins, why do the casinos pay for these displays? Because many players use the previous results to determine what bets to make, even though it won’t help them win. And as long as the players keep making bets, the casino keeps making money.  

Randomness, Probability, and Simulation The imitation of chance behavior, based on a model that accurately reflects the situation, is called a simulation. Randomness, Probability, and Simulation Performing a Simulation State: What is the question of interest about some chance process? Plan: Describe how to use a chance device to imitate one repetition of the process. Explain clearly how to identify the outcomes of the chance process and what variable to measure. Do: Perform many repetitions of the simulation. Conclude: Use the results of your simulation to answer the question of interest. We can use physical devices, random numbers (e.g. Table D), and technology to perform simulations.

Randomness, Probability, and Simulation Whose Book is This? State: When four students mix up their AP Stats books, what is the probability that when each student randomly chooses a book, he doesn’t end up with his own? Plan: On four equally sized slips of paper, write the numbers 1, 2, 3, 4. Shuffle the papers and lay them down one at a time in a row. If the number on the paper matches its position in the row (e.g., paper 2 ends up in the second position), this represents a student choosing his own book. Count the number of students who get the correct book. Do: Do this process many times, recording the number of students who get the correct book in each trial. Randomness, Probability, and Simulation Conclude: Out of 30 total repetitions, there were 11 times when none of the students ended up with their own book. So the estimated probability is 11/30 = 36.7%.

Example: Golden Ticket Parking Lottery Read the example on page 290. What is the probability that a fair lottery would result in two winners from the AP Statistics class? We’ll use Table D to simulate choosing the golden ticket lottery winners. Label theStats students 01-28 and the rest of the students 29-95. After finding two winners, we will record whether both winners were members of the Statistics class. Reading across row 139 in Table D, look at pairs of digits until you see two different labels from 01-95. Record whether or not both winners are members of the AP Statistics Class. Students Labels AP Statistics Class 01-28 Other 29-95 Skip numbers from 96-00 55 | 58 89 | 94 04 | 70 70 | 84 10|98|43 56 | 35 69 | 34 48 | 39 45 | 17 X | X ✓ | X ✓|Sk|X X | ✓ No 19 | 12 97|51|32 58 | 13 04 | 84 51 | 44 72 | 32 18 | 19 40|00|36 00|24|28 ✓ | ✓ Sk|X|X X | ✓ ✓ | X X | X X|Sk|X Sk|✓|✓ Yes No Based on 18 repetitions of our simulation, both winners came from the AP Statistics class 3 times, so the probability is estimated as 16.67%.

Example: NASCAR Cards and Cereal Boxes Read the example on page 291. What is the probability that it will take 23 or more boxes to get a full set of 5 NASCAR collectible cards? Driver Label Jeff Gordon 1 Dale Earnhardt, Jr. 2 Tony Stewart 3 Danica Patrick 4 Jimmie Johnson 5 Use randInt(1,5) to simulate buying one box of cereal and looking at which card is inside. Keep pressing Enter until we get all five of the labels from 1 to 5. Record the number of boxes we had to open. 3 5 2 1 5 2 3 5 4 9 boxes 4 3 5 3 5 1 1 1 5 3 1 5 4 5 2 15 boxes 5 5 5 2 4 1 2 1 5 3 10 boxes We never had to buy more than 22 boxes to get the full set of cards in 50 repetitions of our simulation. Our estimate of the probability that it takes 23 or more boxes to get a full set is roughly 0.

Section 5.1 Randomness, Probability, and Simulation Summary In this section, we learned that… A chance process has outcomes that we cannot predict but have a regular distribution in many distributions. The law of large numbers says the proportion of times that a particular outcome occurs in many repetitions will approach a single number. The long-term relative frequency of a chance outcome is its probability between 0 (never occurs) and 1 (always occurs). Short-run regularity and the law of averages are myths of probability. A simulation is an imitation of chance behavior.

Looking Ahead… In the next Section… We’ll learn how to calculate probabilities using probability rules. We’ll learn about Probability models Basic rules of probability Two-way tables and probability Venn diagrams and probability In the next Section…