Understanding Randomness Chapter 11 Understanding Randomness

What is the most important aspect of randomness? It must be fair. How is this possible? 1) Nobody can guess the outcome before it happens. 2) When we want things to be fair, usually some underlying set of outcomes will be equally likely.

Why Be Random? Example: Statisticians use randomness as a tool. Pick “heads” or “tails.” Flip a fair coin. Does the outcome match your choice? Did you know before flipping the coin whether or not it would match? Statisticians use randomness as a tool. But, truly random values are surprisingly hard to get…

It’s Not Easy Being Random

It’s Not Easy Being Random (cont.) How should we generate random numbers? Pros/Cons? Humans – Statisticians use randomness as a tool. In fact, without randomness we couldn’t do most of statistics. Computers – popular way to generate random numbers. Computers do much better than humans but can’t generate truly random numbers, they are pseudorandom. Random Tables – pseudorandom; appendix G Other – several internet sites can generate truly random digits.

Graphing Calculator - TI Tips (p.263) You have to seed calculator to start at a random place (example) Random Integer Generator (example)

Numbers can be read across. Random digit table Numbers can be read vertically. The following is part of the random digit table found on page 847 of your textbook: Row 1 4 5 1 8 5 0 3 3 7 1 2 4 2 5 5 8 0 4 5 7 0 3 8 9 9 3 4 3 5 0 6 3 Numbers can be read diagonally. each entry is equally likely to be any of the 10 digits digits are independent of each other

Ignore. Ignore. Ignore. Ignore. Suppose your population consisted of these 20 people: 1) Aidan 6) Fred 11) Kathy 16) Paul 2) Bob 7) Gloria 12) Lori 17) Shawnie 3) Chico 8) Hannah 13) Matthew 18) Tracy 4) Doug 9) Israel 14) Nan 19) Uncle Sam 5) Edward 10) Jung 15) Opus 20) Vernon Use the following random digits to select a sample of five from these people. We will need to use double digit random numbers, ignoring any number greater than 20. Start with Row 1 and read across. 1) Aidan 13) Matthew 18) Tracy 5) Edward 15) Opus Ignore. Ignore. Ignore. Ignore. Stop when five people are selected. So my sample would consist of : Aidan, Edward, Matthew, Opus, and Tracy Row 1 4 5 1 8 0 5 1 3 7 1 2 0 1 5 5 8 0 1 5 7 0 3 8 9 9 3 4 3 5 0 6 3

A Simulation Simulation - consists of a collection of things that happen at random. Component - the most basic event of a simulation. Outcomes - Each component has a set of possible outcomes, one of which will occur at random. Trial - the sequence of events we want to investigate. Response Variable - after the trial, we record what happened.

Simulation Steps Identify the component to be repeated. Explain how you will model the outcome. Explain how you will simulate the trial. State clearly what the response variable is. Run several trials. Analyze the response variable. State your conclusion (in the context of the problem, as always).

Shooting foul shots until one is missed Example: Free Throws 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? Step 1: Identify the component to be repeated. Shooting foul shots until one is missed

Free Throws (Cont.) Step 2: Explain how you will model the outcome. Step 3: Explain how you will simulate the trial. Step 4: State clearly what the response variable is. The numbers 0 to 7 will represent a good shot, and 8 or 9 will represent a miss. Use the randInt(0, 9, 30). Why 30? Just to get enough numbers to hopefully encounter a miss. We are interested in the number of hits until she finally misses.

Free Throws (Cont.) Step 5: Run several trials. Step 6: Analyze the response variable. Step 7: State your conclusion In this situation, conducting a simulation is faster and easier than actually shooting free throws. So create a chart and record your findings in 5 trials. Now that we have several trials, we can predict the average number of shots made before a miss. We can now estimate that the average number of baskets made before a miss is = _____

EXAMPLE Continued Trial Number Hits Before Miss 1 6 2 3 4 5 Average = 2.8 hits

What Can Go Wrong? Don’t overstate your case. Always be sure to indicate that future results will not match your simulated results exactly. Model the outcome chances accurately. Run enough trials

ASSIGNMENT A#1/1 p. 266 2, 5, 7, 9, 11, 13