Chapter /5 Simulations / 21

Transcription

1 Chapter /5 Simulations 1

2 Chapter 14 Homework p731 Applying the Concepts p731 1, 2, 5, 6, 7, 8-13, 15, 17, 21 2

3 Objectives: Use simulation to determine probabilities and experimental outcomes. 3

4 Simulation Techniques and the Monte Carlo Method na simulation technique uses a probability experiment to mimic a real-life situation. nthe Monte Carlo method is a simulation technique using random numbers. nthe random numbers can be generated by a computer or via a random number table. We will use the table. 4

5 The Monte Carlo Method 1. List all possible outcomes of the experiment. 2. Determine the probability of each outcome. 3. Set up a correspondence between the outcomes of the experiment and numbers (integers). 4. Select random numbers (integers) from a table and conduct the experiment. 5. Repeat the experiment (trial) and tally the outcomes. 6. Compute any statistics and state the conclusions. 5

6 Example 14-5: Tennis Game Outcomes Using random numbers, simulate the outcomes of a tennis game between Bill and Mike, with the additional condition that Bill is twice as good as Mike. Since Bill is twice as good as Mike, he will win approximately two games for every one Mike wins; hence, the probability that Bill wins will be 2/3, and the probability that Mike wins will be 1/3. 6

7 Example 14-5: Tennis Game Outcomes Using a Random Number Table Random Numbers The random digits 1 through 6 can be used to represent a game Bill wins; the random digits 7, 8, and can be used to represent Mike s wins. The digit 0 is disregarded

8 Example 14-5: Tennis Game Outcomes A set is complete when a player wins 6 Random Numbers games with a margin of victory of 2 games. A match is winning 3 of 5 sets. Outcomes = Bill wins, Mike wins. Simulate a match between Bill and Mike, MMMB xbmbb BB MBB Bill 6-4 BxBBM MBB BB MMBBM Bill 6-3 Bill MBB when Bill is twice as good as Mike. 1-6 Bill, 7-9 Mike, skip Bill takes match 6-4, 6-3,

9 TI-84 We can also simulate a match between Bill and Mike, when Bill is twice as good as Mike, by using the calculator. Since Bill is twice as good as Mike, let 0 represent a win by Mike; 1 and 2 represent wins by Bill. Math PRB 5:randInt(0,2) Enter Each enter generates a random number between 0 and Bill Bill Bill 6-3 Bill takes match 6-2, 6-3, 6-2 9

10 Simulation Experiment A simulation experiment consists of several trials. Each trial consists of 1 or more components. In a simulation, a trial can be a fixed number of components (length) or of varying number of components (length) depending on what we wish to record. In the tennis simulation, each trial ended when we had 6 wins. Each trial (set) was of varying number of components (games). We may wish to record the number of games required to win 6. If we do this repeatedly we can find an estimate of the length of a set by averaging. 10

11 Example of Simulation Experiment A fixed length trial would consist of every trial being of the same number of components. Suppose you are a farmer. You raise chickens for their eggs and you have a hen-house with 10 nests. Because you have been collecting eggs for some time you have found that each hen will lay from 0 to 2 eggs. Based on past experience you estimate that there will be no eggs in 10% of the nests, one egg in 30% of the nests and 2 eggs in the remaining 60%. Conduct a simulation to estimate how many eggs you will collect on each visit to the hen house. 11

12 Example of Simulation Experiment 1. List all possible outcomes of the experiment 0, 1 or 2 eggs. 2. Determine the probability of each outcome. 0 eggs in 10% of nests egg in 30% of nests eggs eggs 2 eggs in 60% of nests eggs 16 eggs 3. Assign numbers We assign 0-9 as follows: 0-0 eggs ,2,3-1 egg ,5,6,7,8,9-2 eggs 6. Based on 4 trials, you can expect about 14 eggs. 12

13 Summary of Simulation Experiment To summarize. A simulation experiment consists of several trials. A trial usually consists of several components. We record the outcome of a trial. A trial can be of fixed length. We record the number of components required to satisfy some criterium (i.e. number of made free-throws in 10 tries). A trial can be of varying length, ending when a condition has been satisfied. We record the number of attempts to complete the trial (number of free-throws required to make 10), or the number of trials to complete some criterium (win the match). 13

14 Example: Cereal Boxes Suppose a cereal manufacturer puts pictures of athletes on cards in their boxes of cereal. The pictures are of Tiger Woods, Serena Williams, and David Beckham. The manufacturer claims that 20% of the boxes contain Tiger Woods, 30% David Beckham, and the rest Serena Williams. You want all three pictures. How many boxes of cereal do you expect to need to purchase to obtain all three athletes? Run a simulation to obtain your answer. 14

15 Example: Cereal Boxes You want all three pictures (outcomes - Tiger, Beckham, Serena). How many boxes of cereal do you expect to need to purchase to obtain all three athletes? Random Numbers 20% of the boxes contain Tiger Woods, % David Beckham, and the rest Serena Williams , 1 - Tiger , 3, 4 - Beckham , 6, 7, 8, 9 - Serena

16 Example: Cereal Boxes How many boxes of cereal do you expect Trial Boxes Trial Boxes to need to purchase to obtain all three athletes? Random Numbers , Tiger , 3, Beckham , 6, 7, 8, Serena

17 Example: Cereal Boxes With a simulation of 24 trials, we can now calculate an expected number of boxes you will need to purchase to get all three athletes. E (b ) = 24 i =1 b i trials E (b ) = = You can expect to buy about 6-7 boxes to get all three athletes. Trial Boxes Trial Boxes

18 How to Answer Note that the way the question is worded indicates how you should respond. The previous example asked how many you would expect to buy. That suggests a mean or average number. Had we been asked how many will you have to buy to ensure you get all three cards? we would probably respond with the maximum number we found, and respond with we may need to purchase 16 boxes to be certain of obtaining all three cards. 18

19 Example Earlier we ran a simulation using beads to investigate the question of gender bias in selecting pilots. We will duplicate that simulation using a random number table. Assign appropriate numbers and run 5 trials for selecting the pilots. Place your results on a dot plot on the board. Compare the results with what you found using the beads. 19

20 Airline Discrimination An airline has just finished training 25 junior pilots 15 male and 10 female to become captains. Unfortunately, only eight captain positions are available right now. Airline managers announce that they will use a lottery process to determine which pilots will fill the available positions. The names of all 25 pilots will be written on identical slips of paper, placed in a hat, mixed thoroughly, and drawn out one at a time until all eight captains have been identified. A day later, managers announce the results of the lottery. Of the 8 captains chosen, 5 are female and only 3 are male. Some of the male pilots who were not selected suspect that the lottery was not carried out fairly. One of the pilots knows that you are taking a statistics class, and comes to you for advice. You offer to consult with your classmates and get back to him. 20

21 Airline Discrimination 0-5 Male Pilots 6-9 Female Pilots fmmmm mmm f f f f 1 female 4 females f Number of females selected Trial 1-1 Trial 6-2 Trial 2-4 Trial f 2 females f ff f 4 females f f 2 females f f f 3 females females f f ff f f f f 7 females f fff f 5 females f f f f 3 females Trial 3-2 Trial Trial 4-3 Trial 5-5 Trial 9-2 Trial