AP STATISTICSObjective: Understanding Randomness

Do Now: Take out any completed contracts, personal profiles, as well as your written design study. HW: Read section 5.3 in text, complete assigned problems from worksheet.warmup: Explain what you think it means for something to be random. Can you think of some real life examples?

Randomness

• In games, randomness usually means things will turn out fair.

• No one will be able to guess the outcome before it happens.

• Usually some set of underlying outcomes will be equally likely.

• Truly random values are hard to get….

• In a minute, I’m going to put up a new slide with some numbers on it. Look at the numbers quickly and then pick a number at random.

• Ready?

• Go.

Did you pick 3?

• Almost 75% of people pick 3.

• About 20% pick either 2 or 4.

• If you picked 1, well, consider yourself a little different.

• Random doesn’t seem so very random after all.

www.dilbert.com

RANDOM NUMBERS

• Are computers random? • They are actually pseudorandom - meaning they

are generated in a fixed sequence. The sequence will eventually repeat itself.

• What about random number tables?• They are okay • Can you think of other ways to generate random

numbers?• Roll dice, spin a spinner, toss a coin, tear up slips

of paper, label them with numbers and place them in a hat, etc.

If someone handed you

a coin and said that it

was biased towards

landing heads, would

you believe them?

What would you do to

verify their claim?

Imagine tossing the coin 100 times

Would 54 out of 100 tosses convince you?

What about 60 / 100?

What about 95 / 100?

What about 80 / 100?

What about 70 / 100?

What would it take to convince you?

MATH-PRB- 5: randInt (min, max, # of trials)

• 0 = represents tails• 1 = represents heads• randInt (0, 1, 5) simulates tossing 5 coins. • Did anyone get all heads or all tails?• sum(randint(0,1, 100))• Since 1 = heads, getting the sum of 100 trials is

the same as counting up how many heads came up.

• Suppose a basketball player has an 80% free throw success rate.

• How can we use random numbers to simulate whether or not she makes a foul shot?

• How many shots might she be able to make in a row without missing?

AP STATISTICS

Do Now: Take out notebook, graphing

calculator, random number table, and blue

packet from last class. From last class:

Suppose a basketball player has an 80%

free throw success rate.

How many shots might she be able to make

in a row without missing?

Write down how you might simulate this…

Simulation

• Component – • taking one shot• get a random digit (0-9)• let 0-7 = a good shot and 8 or 9 = miss• Trials: look at a series of random digits until we an 8 or 9• Response Variable – • Count the number of shots made before the miss• What do we do with all this information?• Find a Statistic -• Find the mean number of shots made

• How would our simulation procedure change if the success rate was only 72%?

• What about if we wanted to know how many shots she might make out of 5 chances?

• What about if we wanted to know her chances of hitting both shots when she goes to the line to shoot two?

Practical Randomness• Suppose a cereal

manufacturer puts pictures of famous athletes on cards in boxes of cereal in the hope of boosting sales. The manufacturer announces that 20% of the boxes contain a picture of Tiger Woods, 30% a picture of Lance Armstrong, and the rest a picture of Serena Williams. You want all three pictures. How many boxes of cereal should you buy in order to get the complete set?

• How do we go about answering a question like this?

• We could buy hundreds of boxes of cereal and see what happens.

• OR we could use a random model (here we are assuming that the pictures are randomly placed in the boxes and that the boxes are distributed randomly to stores around the country).

• Let’s use the random digits 0,1,2,…,9.• 20% of the digits will represent Tiger Woods,

30% Lance, and 50% Serena.

A Simulation

• Simulation – consists of a sequence of random outcomes that model a situation. In this sequence, the most basic event is called a component of the simulation (picking a box of cereal). Each component has a set of possible outcomes, one of which will occur at random (the type of card in the box).

Simulation Steps

1) Identify the component to be repeated.2) Explain how you will model the outcome. The digits 0 – 9 are equally likely to

occur.

20 % Tiger 0, 1 = Woods30% Lance 2, 3, 4 = Armstrong

50 % Serena 5, 6, 7, 8, 9 = Williams

Simulation Steps

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 are pretending to open boxes of cereal until we have one of each athlete.

You could use your calculator to generate some random numbers.

If you get a sequence 29240, that means the first cereal box you opened gave you Lance (2), then Serena (9), then Lance twice more (2, 4), and finally Tiger (0). Since we got all three athletes, our first trial of the simulation is complete.

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 the sample trial, the response value is 5 boxes.

5) Run several trials and make a chart to keep track of what happens.

TI TIPS

Replicate this chart in your notebook and fill it in with your own data (using your random number generator on your calculator or a table)Run 5 Trials

TRIAL NUMBER OUTCOMES Y = # of Boxes

1 89064 = Serena, Serena, Tiger, Serena, Lance

5

2 2730 = Lance, Serena, Lance, Tiger

4

3 8645681 = S, S, L…..,T

8

4

5

Analyze the response variable

• Let’s record the number of boxes we had to “buy” to get all three cards.

• What should we do with all this data?

How many trials is enough?

• Do as many as you can.

• On the AP Test, you might not have time to do that many. Do as many as time allows (depending on the type of simulation you must run), but if you only have time for 5, that’s okay. Make note of the fact that the # of trials was small.

Interesting Fact

• Do you know how many riffle shuffles it takes to mix a deck of cards well?