Chapter 11Understanding

Randomness

What is Random? WAC type 1

Name 5 places you use “randomly” generated outcomes.

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

Pick a Number

1 2 3 4only 5% pick 1

20% pick either 2 or 4

75% pick 3

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

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

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?

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

A Simulation: consists of a sequence of random outcomes that model a situation

Vocabulary 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

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

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

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

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.

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

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?

Using the Calculator

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.

Homework pg 267 # 1 – 5, 7, 9 - 13

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

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

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.

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)

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.

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)