Chapter 5 - Probability Section 1: Randomness, Probability, and Simulation By the end of this section you will be able to: 1) interpret probability as a long-run relative frequency, and 2) Use simulation to model chance behavior. The Idea of Probability (5.1.1) In football, a coin toss helps determine which team gets the ball first. Why do the rules of football require a coin toss? Because tossing a coin seems a fair way to decide. That s one reason why statisticians recommend random samples and randomized experiments. They avoid bias by letting chance decide who gets selected or who receives which treatment. A big fact emerges when we watch coin tosses or the results of random sampling and random assignment closely: chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run. This remarkable fact is the basis for the idea of probability. We might suspect that a coin has probability 0.5 of coming up heads just because the coin has two sides. But we can t be sure. In fact, spinning a penny on a flat surface, rather than tossing the coin, gives heads a probability of about 0.45 rather than 0.5. What about thumbtacks? They also have two ways to land point up or point down but the chance that a tossed thumbtack lands point up isn t 0.5. How do we know 1 of 8
that? From tossing a thumbtack over and over and over again. Probability describes what happens in very many trials, and we must actually observe many tosses of a coin or thumbtack to pin down a probability. The FACT that the proportion of heads in many tosses eventually closes in on 0.5 is guaranteed by the law of large numbers. This important result 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. We call this value the probability. The previous example confirms that the probability of getting a head when we toss a fair coin is 0.5. Probability 0.5 means occurs half the time in a very large number of trials. Outcomes that NEVER occur: probability 0. Outcomes that ALWAYS occur: probability 1. Outcomes that occur half the time: probability 0.5 However, observing an exact probability can NEVER occur. Why? There is always another simulation that can be done that will slightly adjust the previous probability value, even if it is the slightest amount. ***Probability gives us a language to describe the long-term regularity of random behavior.*** The idea of probability is that randomness is predictable in the long run. CYU-292 1. According to the Book of Odds Web site www.bookofodds.com, the probability that a randomly selected U.S. adult usually eats breakfast is 0.61. (a) Explain what probability 0.61 means in this setting. (b) Why doesn t this probability say that if 100 U.S. adults are chosen at random, exactly 61 of them usually eat breakfast? 2. 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 occur once in a while in a long sequence of trials. (d) This outcome will occur more often than not. Myths about Randomness (5.1.2) The idea of probability seems straightforward. It answers the question What would happen if we did this many times? In fact, both the behavior of random phenomena and the idea of probability are a bit subtle. AE-292 Runs in die rolling Roll a die 12 times and record the result of each roll. Which of the following outcomes is more probable? 1 2 3 4 5 6 6 5 4 3 2 1 or 1 5 4 5 2 4 3 3 6 1 2 6 These outcomes are both equally (un)likely, even though the first set of rolls has a more noticeable pattern. 2 of 8
AE-294 - Joe DiMaggio s hitting streak There was an interesting discussion of the hot hand in an article in the New York Times (March 30, 2008) written by Samuel Arbesman and Steven Strogatz (http://www.nytimes.com/2008/03/30/opinion/ 30strogatz.html). In the article, the authors claim that one of the most remarkable streaks in baseball history, Joe DiMaggio s 56-consecutive-game hitting streak, was actually not very remarkable at all. Obviously, it is extremely unlikely for any particular individual to have a hitting streak this long. But, when considering all the players and all the seasons in baseball history, we should expect some very unusual performances every now and then. To investigate, they simulated the performances of every baseball player in every season a total of 10,000 times. In each of those 10,000 simulated histories of baseball, they recorded the longest hitting streak. In about 42% of the trials of the simulation, someone had a hitting streak of at least 56 games in a row, with the longest being an amazing 109 games in a row! Once, at a convention in Las Vegas, one of the authors roamed the gambling floors, watching money disappear into the drop boxes under the tables. You can see some interesting human behavior in a casino. When the shooter in the dice game called craps rolls several winners in a row, some gamblers think she has a hot hand and bet that she will keep on winning. Others say that the law of averages means that she must now lose so that wins and losses will balance out. Believers in the law of averages think that if you toss a coin six times and get TTTTTT, the next toss must be more likely to give a head. It s true that in the long run heads will appear half the time. What is a myth is that future outcomes must make up for an imbalance like six straight tails. Coins and dice have no memories. A coin doesn t know that the first six outcomes were tails, and it can t try to get a head on the next toss to even things out. Of course, things do even out in the long run. That s the law of large numbers in action. After 10,000 tosses, the results of the first six tosses don t matter. They are overwhelmed by the results of the next 9994 tosses. Don t confuse the law of large numbers, which describes the big idea of probability, with the law of averages described here. AE-295 Red is due! In casinos, there is often a large display next to every roulette table showing the outcomes of the previous spins of the wheel. The results of previous spins reveal nothing about the results of future spins, so why do the casinos pay for these displays? The casinos know that many people will be more willing to place a bet if they observe a pattern in the previous outcomes. And as long as people are placing bets, the casino is making money. Simulation (5.1.3) The imitation of chance behavior, based on a model that accurately reflects the situation, is called a simulation. You already have seen an example of simulation: The 1 in 6 wins game that opened this chapter had you roll a die several times to simulate buying 20-ounce sodas and looking under the cap. Stats four step process reviewed. This chapter really begins to develop and require your using the 4-Step Process, especially on any questions outside of multiple choice. The Four Step Process of Performing a Simulation: 1.State - Ask a Question about some chance process 2.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. 3.Do - Perform many repetitions of the simulation. 4.Conclude - Use the results of your simulation to answer the question of interest posed in part 1. 3 of 8
So far, we have used physical devices for our simulations. Using random numbers from Table D or technology is another option, as the following examples illustrate. 4 of 8
AE-297 Stratified sampling 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. ***In the previous example, we could have saved a little time by using randint(1,95) repeatedly instead of Table D (so we wouldn t have to worry about numbers 96 to 00). We ll take this alternate approach in the next example.*** 5 of 8
AE-298 Picking teams 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. CYU - 299 1. Refer to the golden ticket parking lottery example (in your textbook found in Chapter 5; Section 1, on page 296). At the following month s school assembly, the two lucky winners were once again members of the AP Statistics class. This raised suspicions about how the lottery was being conducted. How would you modify the simulation in the example to estimate the probability of getting two winners from the AP Statistics class in back-to-back months just by chance? 2. Refer to the NASCAR and breakfast cereal example (in your textbook Example 7 found on page 297). What if the cereal company decided to make it harder to get some drivers cards than others? For instance, suppose the chance that each card appears in a box of the cereal is Jeff Gordon, 10%; Dale Earnhardt, Jr., 30%; Tony Stewart, 20%; Danica Patrick, 25%; and Jimmie Johnson, 15%. How would you modify the simulation in the example to estimate the chance that a fan would have to buy 23 or more boxes to get the full set? 6 of 8
Chapter 5; Section 1 Homework: 1. The probability of rolling two six-sided dice and having the sum on the two dice equal 7 is 1/6. (a) Interpret this probability. (b) You roll two dice six times. Are you guaranteed to get a sum of 7 once? Explain. 2. To pass the time during a long drive, you and a friend are keeping track of the makes and models of cars that pass by in the other direction. At one point, you realize that among the last 20 cars, there hasn t been a single Ford. (Currently, about 16% of cars sold in America are Fords). Your friend says, The law of averages says that the next car is almost certain to be a Ford. Explain to your friend what he doesn t understand about probability. 7 of 8
3. A bag contains 10 equally-sized tags numbered 0 to 9. You reach in and, without looking, pick 3 tags without replacement. We want to use simulation to estimate the probability that the sum of the 3 numbers is at least 18. Describe the simulation procedure, then use the random number table below to carry out 10 trials of you simulation and estimate the probability. Mark on or above each line of the table so that someone can clearly follow your method. 128 15689 14227 06565 14374 13352 49367 81982 87209 129 36759 58984 68288 22913 18638 54303 00795 08727 130 69051 64817 87174 09517 84534 06489 87201 97245 131 05007 16632 81194 14873 04197 85576 45195 96565 8 of 8