AP STATISTICS LESSON SIMULATING EXPERIMENTS

ESSENTIAL QUESTION: How can simulation be used to solve problems involving chance? Objectives: To simulate problems of chance with the help of a random number table. To simulate problems of chance using the calculator.

Three methods of simulation to answer questions involving chance. 1. Try to estimate the likelihood of a result by actually carrying out the experiment. 2. Develop a probability model and use it to calculate a theoretical answer. 3. Start with a model that, in some fashion, reflects the truth about the experiment, and then develop a procedure for simulating of repetitions of the experiment.

Simulation The imitation of chance behavior, based on a model that accurately reflects the experiment under consideration, is called a simulation. Independent (trials) – One event has no effect or influence over the next (e.g. coin tosses).

Simulation Steps Step 1: State the problem or describe the experiment. Step 2: State the assumptions. Step 3: Assign digits to represent outcomes. Step 4: Simulate many repetitions. Step 5: State your conclusions.

Simulation using the calculator or computer Using the calculator or statistical software to estimate probability through simulation is faster than doing it by hand.

Problem 5.67 Tennis Racquets Professional tennis players bring multiple racquets to each match. They know that high string tension, the force with which they hit the ball and occasional “racquet abuse” are all reasons why racquets break during a match. Brian Lob’s coach tells him that he has a 15% chance of breaking a racquet in any given match. How many matches on average can Brian expect to play until he breaks the racquet and needs to use a backup? Use simulation methods to answer the question. Step 1: State the problem or describe the experiment. Brian has a 15% chance of breaking his racquet in any given match. We want to know how many matches he can play until he breaks his racquet. Step 2: State the assumptions: We can assume that the racquet use in the matches is independent. One match does not effect the other in terms of the racquet We know that he has a 15% chance of breaking the racquet leaving a 85% chance of not breaking the racquet. Step 3: Assign digits to represent outcomes: I will assign 00 to 14 to represent breaking the racquet I will assign to represent not breaking the racquet

Problem 5.67 Continued Step 4: Simulate many repetitions: Using our calculator we can generate one number at a time to decide how many matches it took Brian to break the racquet, record that number, and then repeat the simulation. RandInt(00,99,1) We will repeat the simulation twice each and then compute the average number of matches he will play before his racquet breaks. Step 5: State your conclusions: We can conclude that Brian will play an average of games before he breaks the racquet and has to use his backup. In chapter eight, we will actually Use probability models to calculate the true number of matches Brian will play before he breaks his racquet.

Class Activity (5.3 Simulating Experiments) Part 2 Use your calculator to simulate a couple’s having children until they have a girl or until they have 4 children. Use the simulation to estimate the probability that they will have a girl among their children. First of all, we still have to follow the steps of simulation. Step 1: State the problem or describe the experiment: Step 2: State the assumptions: Step 3: Assign digits to represent outcomes: Step 4: Simulate many repetitions: Step 5: State your conclusions: Now, using our calculator, we were able to do many more repetitions in a timely manor. Therefore our probability is more accurate. A couple will have children until they have a girl or until they have four children. We can assume that the observations are independent of each other (this means that the sex of one child will not effect the sex of the next child). Also, a boy or girl is equally likely to occur. Let 0 represent a boy and 1 represent a girl. Using randInt(0,1,4) we can simulate a couple’s having 4 children. We will do this 10 times each and combine our results to estimate the probability. Taking the number of desired outcomes out of the total number of repetitions, we can compute the probability of having a girl for this couple to be: Estimated probability =