Chapter 5 Understanding Randomness AP Statistics

Why Be Random? Before every volleyball game, we flip a coin to determine which team will decide who serves first. Why do we have to flip a coin? Why can’t we just let the referee decide who serves first? Let’s say that you notice that one official always seems to flip heads. The referee does not seem to be doing anything fishy and doesn’t understand why he keeps getting heads either – so you believe that the referee has an “unfair” (or weighted) coin. In an attempt to find out why he keeps getting heads, the referee gives you the coin to determine if it is…

Why Be Random? How can you determine if the coin is really biased or if it’s a “fair” coin? You’d probably would begin by tossing it several times to see if it gives you an awkward distribution. There once was a Russian czar that wanted to know if a particular coin was fair and he made one of his advisors flip it several thousands of times! How many times is enough? Is two times enough? 10 times? 100? OK, let’s say 100, for arguments sake…then what? If you got 52 heads out of 100 flips, would you consider the coin unfair? Would you consider it fair? What about 53 heads out of 100? What about 58? 60? Well what about 95 out of 100? Ok, so where do we draw the line?

Why Be Random? Let’s take a poll…everyone decide for yourself at what number you would consider the coin to be biased. Would 55 out of 100 be considered biased? Would 70? What’s the number that would convince you? Well what happens when you flip a “fair” coin? Could you see 55 heads out of 100? What about 60 heads out of 100? Is it possible to see 95 heads out of 100? OK, so how do we decide if a coin is fair or unfair? Before we answer this question, let’s talk about randomness

Definition of Random What does it mean to be random? Most people confuse randomness with haphazardness, but these two things are somewhat different. To understand randomness better, let’s define it: We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions. In other words: “Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run.”

Definition of Random A common misinterpretation of randomness is to think that it means equally likely; however, randomness does NOT mean that each outcome is equally likely To further explore this idea consider the following… If 100,000 people purchase a lottery ticket does each person have an equally likely chance of winning? Is this a random event? Yes, each person has the same chance of winning and it is random Since this is random, does each outcome have the same chance of occurring? Definitely NOT! There is a very high chance of losing the lottery, but a very low chance of actually winning!

Why Be Random? (cont.) Using the definition of randomness, consider the following: If you roll a fair six sided die, what number will you expect to land face up? Will everyone in the class get the same number? Is this random? However, in the long run (after a 6000 rolls), how many times would you expect a six to show up? Statisticians don’t think of randomness as the annoying tendency of things to be unpredictable or haphazard. Statisticians use randomness as a tool.

It’s Not Easy Being Random (cont.) There are ways to generate random numbers so that they are both equally likely and truly random. One method is to use a Table of Random Numbers Another method is to use technology The best ways we know to generate data that give a fair and accurate picture of the world rely on randomness, and the ways in which we draw conclusions from those data depend on the randomness, too.

Back to the Case of the Unfair Coin Ok, let’s flip a “fair” coin and determine what kind of results we would get. How many times should we flip the coin to see what we would consider to be “normal” or typical? Let’s flip the coin 100 times each using technology… If we use RandInt(min, max, n), it will generate n number of random integers Use RandInt (0, 1, 1) where 0 represents tails and 1 represents heads.

The Random Number Table

The Random Number Table

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. The sequence of events we want to investigate is called a trial. Each simulated answer to our question, it is called a trial. The basic building block of a simulation is called a component (trials usually involve several components). After the trial, we record what happened—our response variable or outcomes.

4 Steps to Simulation State: What is the question of interest about some chance process? Plan: Describe how to use a chance device to imitate one repetition of the process. Explain clearly how to identify the outcomes of the chance process. Do: Perform many repetitions of the simulation. Conclude: Use the results of your simulation to answer the question of interest.

Example: How Do You Answer… There are some questions that are difficult to answer. Such as: Suppose there is a child that wishes to obtain certain “Star Wars” figures from McDonald’s Happy Meals. McDonalds announces that 10% of the figures are Boba Fett, 30% of the figures are Darth Vader, and 60% of the figures are Luke Skywalker. If the child wants all three figures, how many Happy Meals would the child expect to purchase before he/she get all three?

Simulations Let’s use simulation answer this question. Apply the 4 steps to simulating an experiment: Step 1: State what is the question of interest. In our example, the component is the selection of a cereal box. We want to know how many Happy Meals the child would have to purchase before getting all three toys.

Simulations Apply the 4 steps to simulating an experiment: Step 2: Plan. The digits 0 to 9 are equally likely to occur. Because 10% of the Happy Meals contain Boba Fett, 30% have Darth Vader, and 60% have Luke Skywalker, we will let 0 represent Boba Fett, 1-3 represent Darth Vader, and 4-9 represent Luke Skywalker A trial is the sequence of events that we are pretending will take place. In this case, we want to pretend to buy Happy Meals and indicate the outcome of the figure. (Simulations are cheaper than running actual experiments. Running many trials is critical to an appropriate simulation.)

Let’s “Run” the Simulation (Step 3) Using the calculator, let’s run through the simulation: Type: randInt(0, 9, 1) into your calculator Remember that we have the following assignment: Boba Fett: 0 Darth Vader: 1, 2, and 3 Luke Skywalker: 4, 5, 6, 7, 8, and 9 Keep pressing [Enter] until you get one of each “figure” How many times did it take you to get all 3 “figures”? Repeat the simulation. Step 4: Conclude- How many “Happy Meals” should you expect to buy in order to get all 3 figures?

The Random Number Table In addition to using the calculator, you should be able to use a list of random numbers from a random number table. See if you can simulate the following: If there is a 30% chance of getting the flu, simulate how many people will get the flu if there are 17 people. Use line 103. There is a 61% chance of rain for the month. Simulate how many days it will rain over the next week. Use line 120. There is a 75% chance of failing the PRAXIS exam. Simulate how many of the next 15 people will fail. Use line 114.

To Sum It Up… A simulation model can help us investigate a question for which: many outcomes are possible, we can’t (or don’t want to) collect data, and the mathematical answer is hard to calculate. We base our simulations on random values. Like all models, simulations can provide us with useful insights about the real world.

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.