Chapter 11 Randomness

1. Chapter 11 Randomness

2. Randomness • Random outcomes • Tossing coins • Rolling dice • Spinning spinners • They must be fair

3. Randomness • Random outcomes • Tossing coins • Rolling dice • Spinning spinners • They must be fair • Nobody can guess the outcome before it happens

4. Randomness • Random outcomes • Tossing coins • Rolling dice • Spinning spinners • They must be fair • Nobody can guess the outcome before it happens • Usually some underlying set of outcomes will be equally likely

5. 1 2 3 4

6. Result

7. Result It is not easy to be random!

8. It’s Not Easy Being Random • Computers have become a popular way to generate random numbers. • Even though they often do much better than humans, computers can’t generate truly random numbers either. • Since computers follow programs, the “random” numbers we get from computers are really pseudorandom. • Fortunately, pseudorandom values are good enough for most purposes.

9. Example • A cereal manufacturer puts pictures of famous athletes on cards in boxes of cereal in the hope of boosting sales • 20% of the boxes contain a picture of Tiger woods • 30% for Lance Armstrong • 50% for Serena Williams • Get a lottery ticket when all three pictures are collected

11. How many boxes do we need to buy? • If you are lucky, • If you are very unlucky,

12. How many boxes do we need to buy? • If you are lucky, three boxes • If you are very unlucky, infinitely many

13. How many boxes do we need to buy? • If you are lucky, three boxes • If you are very unlucky, infinitely many • But on average, how many? • Go ahead and buy, then count (too costly) • A cheaper solution • Use a random model • Assume that pictures are randomly placed in the boxes • Assume that the boxes are randomly distributed to stores

14. Practical Randomness • We need an imitation of a real process so we can manipulate and control it. • In short, we are going to simulate reality.

15. Then what? • Use the model to generate random values • Simulate the outcomes to see what happens • We call each time we obtain a simulated answer to our question a trial • How do we generate the outcomes at random?

16. Random numbers • How to generate random numbers? • Computer software • Pseudorandom numbers • Computers follow programs! • The sequence of pseudorandom numbers eventually repeat itself • But virtually indistinguishable from truly random numbers • Books of random numbers • Not an interesting book, perhaps 

18. How do we get random integers in TI-83? • MATH PRB • 5:randInt( • randInt(left,right, #) • This allows repetition of the same integer • randInt (0,9,100) produces 100 random digits • randInt (0,57,3) produces three random integers between 0 and 57 • Remark: The textbook’s comments are wrong. Using this function, you can possibly get (14, 41, 14), which is not suitable for the dorm room example.

19. Back to cereal boxes • How to model the outcome? • 20%, Woods (0,1) • 30% Armstrong (2,3,4) • 50% Williams (5,6,7,8,9) • 0 to 9 are equally likely to occur • How to simulate the trial? • Open cereal boxes till we have one of each picture • Opening one box is the basic building block, called a component of our simulation. • For example, ‘29240’ corresponds to the following outcomes:

20. Back to cereal boxes • How to model the outcome? • 20%, Woods (0,1) • 30% Armstrong (2,3,4) • 50% Williams (5,6,7,8,9) • 0 to 9 are equally likely to occur • How to simulate the trial? • Open cereal boxes till we have one of each picture • Opening one box is the basic building block, called a component of our simulation. • For example, ‘29240’ corresponds to the following outcomes: Armstrong, Williams, Armstrong, Armstrong, Woods

21. Cereal (continue) • Response variable • What we are interested in • How many boxes it takes to get all three pictures • Length of the trial • ‘29240’ corresponds to 5 boxes • Run more trials • 89064: 5 boxes • 2730: 4 boxes • 8645681: 7 boxes • 41219: 5 boxes • 822665388587328580: 18 boxes • How many boxes do we expect to buy? • Take an average • Based on the first 5 trials: the average is 7.8 boxes • To get an objective estimate: run infinitely many trials

22. Cautions • Simulation is different from the reality because • Model may not be 100% precise • Only limited number of trials • Run enough trials before you draw conclusions

23. Simulation Steps • Identify the component to be repeated. • Explain how you will model the component’s outcome. • State clearly what the response variable is. • Explain how you will combine the components into a trial to model the response variable. • Run many trials. • Collect and summarize the results of all the trials. • State your conclusion.

24. Lottery for a dorm room • 57 students participated in a lottery for a particularly desirable dorm room • A triple with a fireplace and private bath in the tower • 20 participants were members of the same varsity team • When all three winners were members of the team, the other students cried foul

25. Is it really a foul? • Whether an all-team outcome could reasonably be expected to happen if every one is equally likely to be selected? • Simulation • Component: selection of a student • 00-56: one number for one student • 00-19: 20 varsity applicants • 20-56: the other 37 applicants • Trial: randomly select three numbers from 00-56 • Note we can’t put the same person in the room twice • If selecting the second person, the first selected person should be excluded from the lottery • Response variable: ‘all varsity’ or not • Draw conclusions by counting how often ‘all varsity’ occurs • The textbook gives 10% (only 10 trials) • If doing infinitely many trials, it should be 3.896104%

26. What Can Go Wrong? • Don’t overstate your case. • Beware of confusing what really happens with what a simulation suggests might happen. • Model outcome chances accurately. • A common mistake in constructing a simulation is to adopt a strategy that may appear to produce the right kind of results. • Run enough trials. • Simulation is cheap and fairly easy to do.