Chapter 11 Understanding Randomness

1. Chapter 11Understanding Randomness

2. What is Random? • WAC type 1 • Name 5 places you use “randomly” generated outcomes.

3. Randomness • What makes something fair? • No one can guess the outcome before it happens • The outcomes are equally as likely • Random outcomes have a lot of structure when viewed in the long run • A coin. • You can not guess what the next outcome will be • There is no benefit to one side • Over many trials you will see 50% heads

4. Pick a Number 1 2 3 4 only 5% pick 1 20% pick either 2 or 4 75% pick 3

5. Generating Random Numbers • People • not reliable • Computers • not actually random; a computer uses algorithms to generate random numbers. • we call them pseudorandom (generated by a fixed sequence) • Books • You can buy books of truly random numbers • they use methods like timing of the decay of a radioactive element or changes in lava lamps

6. Shuffling Cards • It was discovered by statisticians that you need to shuffle a deck 7 times to get a truly random order to the cards. • Fewer times leaves order in the deck • More than 7 times does little good

7. Practical Randomness • Suppose a cereal company puts pictures of athletes on cards in boxes of cereal in hopes to boost sales. The company announces that 20% of boxes contain a picture of Tiger Woods, 30% a picture of Lance Armstrong, and the rest (50%) will have a picture of Serena Williams. • You want all three cards. How many boxes of cereal do you expect to buy to get all three cards?

8. Set Up a Model • We will use the digits 0, 1, 2, 3,….9 • Assuming they are equally likely to come up and we don’t know which one will be next • Tiger Woods is in 20% of the boxes so he should be represented by 20% of the digits we use • Lance Armstrong is in 30% of the boxes so he should be represented by 30% of the digits we use • Selena Williams is in 50% of the boxes so she should be represented by 50% of the digits we use

9. A Simulation: consists of a sequence of random outcomes that model a situation Vocabulary Example Buying a box of cereal The card that is in the box Getting all three cards How many boxes did we need to buy to get all three cards • Component: the most basic part of a simulation • Outcomes: each component has a set of possible outcomes • Trial: the sequence of events we want to investigate • Response Variable: what happened

10. Steps To Making A Simulation • 1. Identify the component to be repeated • In this case, our component is the selection of a box of cereal. • 2. Explain how you will model the outcomes. • The digits 0 to 9 are equally likely to occur. Because 20% of the boxes contain Tiger’s picture, we’ll use 2 of the ten digits to represent that outcome. Three of the ten digits can model the 30% of boxes that contains Lance cards, and the remaining 5 digits can represent the 50% of boxes with Serena. • 0, 1 = Tiger • 2, 3, 4 = Lance • 5, 6, 7, 8, 9 = Serena

11. Steps Cont. • 3. Explain how you will simulate the trial. • A trial is the sequence of events that we are pretending will take place. In this case we want to pretend to open cereal boxes until we have one of each picture. We do this by looking at each random number and indicating what outcome it represents. We continue until we have encountered all three pictures • Example: the sequence 29240 • 2 – Lance • 9 – Serena • 2 – Lance • 4 – Lance • 0 – Tiger • Since we got all three pictures that is the end of a trial

12. Steps Cont • 4. State clearly what the response variable is • What are we interested in? We want to know how many boxes it takes to get all three pictures. This is the response variable. In this sample trial here, the response variable is 5 boxes. • 5. Run several trials • A simulation is cheaper than really buying cereal, and the more trials you perform, the better.

13. Last Steps • 6. Analyze the response variable. • We wanted to know how many boxes we might expect to buy to get all three cards. To answer the question, we need to analyze the response variable. • Mean: 7.8 boxes for the first 5 trials • 7. State your conclusion. • Based on our simulation, we estimate that customers who want the complete set of sports star pictures will by an average of 7.8 boxes of cereal. • We only ran 5 trials which is not enough (but for example sake it works) • You should run at least 20 trials when working by hand • You should run a few hundred when using a computer

14. Checking In • The baseball World Series consist of up to seven games. The first two are played at one team’s home park, the next three are played at the other team’s park, and the final two (if needed) are played back at the first park. Records over the past century show that there is a home field advantage; the home team has about a 55% change of winning. Is it an advantage to play the first two games on your home field? Or would you rather have the three middle games at home with the opportunity to win it all? In other words, do the two teams have an equal chance to win the four games needed to win the Series? • What is the component to be repeated? • How will you model the outcome? • How will you simulate the trial? • What is the response variable? • How will you analyze the response variable?

15. Using the Calculator

16. Example (#11 pg 267) • Continuing to use the cereal scenario. Suppose you buy five boxes of cereal. Estimate the probability that you end up with a complete set of pictures. Your simulation should have at least 20 runs.

17. Homework • pg 267 # 1 – 5, 7, 9 -13

18. Homework pg 266 • 1. Yes. We can not guess what will come next and both sides are equally as likely. • 2. That the outcome cannot be predicted and that all #s are equally as likely to come up. • 3. A machine pops up numbered balls. It is random if you can not guess what number will be next and if each ball has the same chance of being pulled. • 4. example answers: pulling a card, spinning a spinner, rolling dice

19. Homework pg 266 (cont) • 5. a) they are not equally likely b) the even/odd simulation assumes that the shooter is equally like to hit a shot. (probably not the case) c) example: 1, 1, 1, 1, 5 means that 4 aces where drawn and then a 5. After the first ace is drawn the odds to get the next one are going to be different

20. Homework pg 266 • 7. The conclusion should say the simulation suggests the average number of people is 3.2 • 9. • a) The component is one voter voting. An outcome is a vote for our candidate or not. Use 2 random digits from 00 – 99. Giving 00 – 54 to a vote for your candidate and 55 – 99 to be a vote for the underdog. • b) The trial is 100 votes. So simulate 100 random 2 digit numbers, then count up the votes and determine the winner. • c) The response variable is whether the underdog wins or not.

21. Homework pg 266 • 10. • a)The component is picking a single card. An outcome is the suit and the number of the card. Use 01 – 52 for 52 different cards. (ignore 00, 53 – 99) • b) A trial is a single 5-card hand. Use 5 sets of random numbers, ignoring repeats and non-defined numbers. • c) Response variable: record if the hand has a pair, three of a kind, or neither • Example digit assignment: 01 – A (h), 02 – 2 (h), 03 – 3(h), ...12 – Q(h), 13 – K (h) 14 – A (d), 15 – 2 (d), 16 – 3 (d), …25 – Q (d), 26 – K (d)

22. Homework pg 266 (c0nt) • 11. on the calculator: randint(0, 9, 5) • example trials: • 93288 • 45950 • 08848 • 80189 • 64529 • 98393 • 24038 • The simulation suggests you will get all three cards when you buy 5 boxes about 30% of the time.

23. Homework pg 266 (cont) • 12. trial: getting a Tiger Woods card response variables: 9, 1, 3, 7, 11, 4, 1, 6, 1, 5 • On average you would have to buy 4.8 boxes to get a Tiger Woods card. • 13. component: each multiple choice question outcome: right or wrong trial: answering 6 questions response variable: how many times you get all right • calc: randint(0, 9, 6)