The Practice of Statistics Third Edition Chapter 6: Probability and Simulation: The Study of Randomness 6.1 Simulation Copyright © 2008 by W. H. Freeman & Company Daniel S. Yates

Essential Question What is a simulation? What are the five steps involved in a simulation? What are independent trials? Given a probability problem, how do you use simulation to get the estimated probability?

What is Probability? Probability is the branch of mathematics that describes the pattern of chance outcomes. Based on The Big Idea: “Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run.” Attempts to answer: “What would happen if we did this many times?”

Definition of Random 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.

Definition of Probability The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. That is, probability is long-term frequency.

Three Methods for Estimating the Chances (Probability) of an Event Occurring Observe the random phenomenon many times and calculate the relative frequency of the results. Develop a probability model and calculate the theoretical answer. Develop a simulation (model with a plan for imitating the a number of repetitions of the random phenomenon).

Simulation Steps Step 1: State the Problem or describe the random event. Step 2: State the assumptions. Step 3: Assign digits to represent outcomes. Step 4: Simulate man y repetitions. Step 5: State your conclusions.

Independent Event The outcome of one trial (event) does not influence or change the outcome of another trial.

Example Look at example 6.3 on page 394

Example 2 Cory rolls a die 30 times. How often does a number of 2 or less appear?

Example 2 Continued Step 1: State the problem. Roll a die for 30 times. What is the probability of rolling a 2 or less? Step 2: Assumptions. The outcomes are independent. The probability of rolling a number is equally likely for any of the six number on the die. Step 3: Assign digits to represent outcome. –Use 1 through 6 to represent each number on the die. Step 4: Simulate 30 repetitions.

Example 2 Step 4 Repetitions Math---PRB---randInt(1,6,30)--Sto— L1--Enter

Step 4 Repetitions Categorize the results. L1– 2 nd Test--≤--2 –Sto--L2--Enter

Step 4 Repetition 2 nd List—MATH—Sum(L2)—Enter.

Step 5 Conclusion The estimated probability for rolling a 2 or less in thirty rolls = 11/30 = Is this what you expected? Why wasn’t it exactly 10. What would happen if Cory “rolled” 300 times?

Law of Large Numbers The long-run relative frequency of repeated independent events settles down to the true probability as the number of trials increases. Applet – coin toss

Example 3 Fifty-seven students participated in a lottery for a particularly desirable dorm room, a triple with a private bath. Twenty of the participants were members of the same varsity team. When all three winners were members of the team, the other students cried foul. Use a simulation to determine whether an all-team outcome could reasonably be expected to happen if everyone had a fair shot at the room.(Stats Modeling the World, page 262)

Example 3 Continued Step 1 : State the problem. What is the probability of having an all-team outcome? Step 2: Assumptions. –Each individual outcome is independent. –Each of the 57 participants have an equal chance of being selected. Step 3: Assign digits to each outcome. Use a calculator to do the simulation. –0 – 19 represent the 20 varsity team members. –20 – 56 represent the other students.

Example 3 Step 4 Simulate 10 Repetitions TrialOutcomeAll Varsity 4, 56, 18V, N, VNo 29, 23, 13N, N, VNo 6, 55, 12V, N, VNo 9, 54, 22V, N, NNo 22, 36, 13N, N, VNo 32, 15, 42N, V, NNo 20, 42, 23N, N, NNo 31, 8, 8N, V, VNo 46, 24, 34N, N, NNo 23, 8, 22N, V, NNo

Example 3 Step 5 Conclusion The estimated probability of having an all team outcome equals 0/10 = 0.0. It appears that the claim of foul may be true because the probability of an all team outcome is very small.