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

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

3. 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.

4. + Check Your Understanding Example 1 Probability is a measure of how likely an outcome is to occur. Match one of the probabilities that follow with each statement. Be prepared to defend your answer. {0, 0.01, 0.3, 0.6, 0.99, 1} a) This outcome is impossible. It can never occur. b) This outcome is certain. It will occur on every trial. c) This outcome is very unlikely, but it will ocurr once in a while in a long sequence of trials. d) This outcome will occur more often than not.

5. + Solution 0 – never occur 1 – always occur 0.01 – approximately 1% 0.6 – approximately 60% will occur more than not occur, whereas.99 means it happens ALMOST every time.

6. + Example 2 4 students are studying together. While they are getting a snack in the kitchen, someone’s younger brother comes in and mixes up all the textbooks. Each student takes a book at random. The graphs below show the short-run and long-run behavior of the proportion of trials in which there are no matches when four 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.

7. + 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 aspecific outcome occurs approaches a single value.

8. + Example 3 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. What is the minimum amount the company should charge for the extended warranty?

9. + Solution 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.

10. + Myths about Randomness The idea about probability is that randomness is predictable in the long run. Unfortunately, our intuition about randomness tries to tell us that random phenomena should also be predication in the short run. When they aren’t, we look for some explanation other than chance variation.

12. + Randomness, Probability, and Simulation Example 4 Roll a die 12 times and record the result of each roll. Which of the following outcomes is more probable? 123456654321or 154524336126

13. + Randomness, Probability, and Simulation Example 4 Solution These outcomes are both equally (un)likely, even though the first set of rolls has a more noticeable pattern.

14. + Example 5 Suppose that a basketball announcer suggests a certain player is streaky. That is, the announcer believes that if the player makes a shot, then the player is more likely to make his next shot. As evidence, the announcer points to a recent game where the player took 30 shots and had a streak of 7 made shots in a row. Is this convincing evidence of streakiness, or could it have occurred simply by chance? Assuming that this player makes 50% of his shots and the results of a shot don’t depend on previous shots, how likely is it for the player to have a streak of 7 or more made shots in a row? Using your calculator, let 1 represent a made shot and 0 represent a missed shot. Calculate a random integer (either 0 or 1) 30 times, writing down the outcome after each calculation. Record the length of the longest streak of made shots. Combine your results with those of your classmates. In what proportion of the trials did the player have a streak of at least 7 in a row?

15. + STATE How likely is it for the player to have a streak of 7 or more shots in a row?

16. + PLAN Use the random number generator on the calculator to generate 30 random shots. 0 represents a missed shot. 1 represents a shot made. Record the outcome of each trial. Record the longest streaks on a dotplot.

17. + DO Perform this twice. _________________________________________________________________________________ 2 3 4 5 6 7 8 9 10

18. + CONCLUDE The player had a streak of 7 or more made shots in ____ of the ____ simulated games (_____%).

19. + Randomness, Probability, and Simulation 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? Example 6

20. + Solution 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.

21. + Section 5.1 Randomness, Probability, and Simulation After this section, you should be able to… DESIGN and PERFORM simulations Learning Objectives

22. + Simulation The imitation of chance behavior, based on a model that accurately reflects the situation, is called a simulation. Randomness, Probability, and 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. Performing a Simulation We can use physical devices, random numbers (e.g. Table D), and technology to perform simulations.

23. + IMPORTANT When making conclusions, be careful to say that there is either convincing evidence (or there isn’t convincing evidence) to support a particular claim. DO NOT say that a claim is definitely true or that the evidence proves that a claim is incorrect.

24. + Example 1 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 the papers 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, because I have already mixed up the names, I don’t want to have to separate them all again. Instead, I will select names one at a time from the bag 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 10 or more names to get 4 seniors and 2 juniors.

25. + STATE What is the probability that it takes 10 or more selections to get 4 seniors and 2 juniors?

26. + PLAN Using pairs of digits from Table D, we’ll label the 60 seniors 01– 60 and the 30 juniors 61–90. Numbers 00 and 91–99 will be skipped. Moving left to right across a row, we’ll look at pairs of digits until we have 4 different labels from 01–60 and 2 different labels from 61–90. Then we will count how many different labels from 01–90 we looked at.

27. + DO Here is an example of one repetition, using line 101 from Table D: 19 (senior) 22 (senior) 39 (senior) 50 (senior) 34 (senior) 05 (senior) 75 (junior) 62 (junior) In this trial, it took exactly 8 selections to get at least 4 seniors and at least 2 juniors. Here are the results of 50 trials:

28. + CONCLUDE In the simulation, 11 of the 50 trials required 10 or more selections to get 4 seniors and 2 juniors, so the probability that it takes 10 or more selections is approximately 0.22.

29. + Example 2 At their annual picnic, 18 students in the mathematics/statistics department at a university decide to play a softball game. Twelve of the 18 students are math majors and 6 are stats majors. To divide into two teams of 9, one of the professors put all the players’ names into a hat and drew out 9 players to form one team, with the remaining 9 players forming the other team. The players were surprised when one team was made up entirely of math majors. Is it possible that the names weren’t adequately mixed in the hat, or could this have happened by chance? Design and carry out a simulation to help answer this question.

30. + STATE What is the probability that when randomly assigning 12 Math majors and 6 Stats majors to two teams that there will be one team with all Math majors?

31. + PLAN Using pairs of digits from Table D, we’ll label the 12 math majors 01–12 and the 6 stats majors 13–18. Number 00 will be skipped. Moving left to right across a row, we’ll look at pairs of digits until we have a team of 9.Then we will count how many different labels from 01–18 we looked at.

32. + DO Here is an example of one trial: Team A: MMMSMMMMM (8 Math majors) Team B: MSMMSSSSM (4 Math majors) Since the team with the most Math majors had 8, we will record the value 8 for this trial. Here are the results of 30 trials:

33. + CONCLUDE Since only 1 trial in 30 resulted in a team with all Math majors, the probability is only approximately 0.033. Since getting a team of all Math majors is unlikely, we can conclude that the names were probably not shuffled very well in the hat.

34. +

35. + 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? StudentsLabels AP Statistics Class01-28 Other29-95 Skip numbers from 96-00 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. 55 | 5889 | 9404 | 7070 | 8410|98|4356 | 3569 | 3448 | 3945 | 17 X | X ✓ | X X | X ✓ |Sk|X X | X X | ✓ No 19 | 1297|51|3258 | 1304 | 8451 | 4472 | 3218 | 1940|00|3600|24|28 ✓ | ✓ Sk|X|X X | ✓✓ | X X | X ✓ | ✓ X|Sk|X Sk| ✓ | ✓ YesNo YesNoYes 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%.

36. + 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? DriverLabel Jeff Gordon1 Dale Earnhardt, Jr.2 Tony Stewart3 Danica Patrick4 Jimmie Johnson5 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. 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. 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

37. + Section 5.1 Randomness, Probability, and Simulation 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. Summary

38. + Looking Ahead… 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…