Section Using Simulation to Estimate Probabilities Objectives: 1.Learn to design and interpret simulations of probabilistic situations

Section Using Simulation to Estimate Probabilities Generating Random Integers Using a Table of Random Digits Using the TI-83/84 Calculator

Section Using Simulation to Estimate Probabilities Using a Table of Random Digits (Table D - p 828) Generating Random Integers from 0 to 99 Each row consists of 10 columns of five digits each. Ignore the spaces in each row when selecting random digits. Choose a row at random (OK to use the first row). Mark off two-digit numbers until you have the amount you need.

Section Using Simulation to Estimate Probabilities Using a Table of Random Digits (Table D - p 828) On the AP Statistics Exam, students typically must do simulations using specific random digits that are included in the question. This is necessary so the readers can verify that the the student did the simulation correctly.

Section Using Simulation to Estimate Probabilities Generating Random Integers from 0 to 99 Using the TI-83/84 Calculator: Key strokes: MATH => PRB 5 0,99 ENTER Each time you press ENTER you get a new random integer between 0 and 99:

Section Using Simulation to Estimate Probabilities The Steps in a Simulation That Uses Random Digits 1.Assumptions: State the assumptions you are making about how the real life situation works. Include any doubts you might have about the validity of your assumptions. 2.Model: Describe how you will use random digits to conduct one run of a simulation of the situation Make a table that shows how you will assign a digit (or a group of digits) to represent each possible outcome. (You can disregard some digits) Explain how you will use the digits to model the real-life situation. Tell what constitutes a single run and what summary statistic you will record.

Section Using Simulation to Estimate Probabilities The Steps in a Simulation That Uses Random Digits 3.Repetition: Run the simulation a large number of times, recording the results in a frequency table. You can stop when the distribution doesn’t change to any significant degree when new results are added. (On a quiz or test, you will be asked to do a few runs, about 10 or so.) 4.Conclusion: Write a conclusion in the context of the situation. Be sure to say you have an estimated probability.

Section Using Simulation to Estimate Probabilities P10. How would you use a table of random digits to conduct one run of a simulation of each situation? 1.There are eight workers, ages 27, 29, 31, 34, 34, 35, 42, and 47. Three are to be chosen at random for layoff. 2.There are eleven workers, ages 27, 29, 31, 34, 34, 35, 42, 42, 42, 46, and 47. Four are to be chosen at random for layoff.

Section Using Simulation to Estimate Probabilities P10a. There are eight workers, ages 27, 29, 31, 34, 34, 35, 42, and 47. Three are to be chosen at random for layoff. Assumptions: You are assuming that each of the eight workers has the same chance of being laid off and that the workers to be laid off are selected at random without replacement.

Section Using Simulation to Estimate Probabilities Model: Assign each worker a random digit as shown: OutcomeDigit Assigned The worker aged 271 The worker aged 292 The worker aged 313 The first worker aged 344 The second worker aged 345 The worker aged 356 The worker aged 427 The worker aged 478

Section Using Simulation to Estimate Probabilities Model: Start at a random place in a table of random digits. The next three digits represent the workers selected to be laid off. If a 9 or 0 appears, ignore it and go to the next digit. Also, because the same person can’t be laid off twice, if a digit repeats, ignore it and go to the next digit.

Section Using Simulation to Estimate Probabilities P10b. There are eleven workers, ages 27, 29, 31, 34, 34, 35, 42, 42, 42, 46, and 47. Four are to be chosen at random for layoff. Assumptions: You are assuming that each of the eleven workers has the same chance of being laid off and that the workers to be laid off are selected at random without replacement.

Section Using Simulation to Estimate Probabilities Model: Assign each worker a random digit as shown: OutcomeDigit AssignedSet of Digits Assigned The worker aged The worker aged The worker aged The first worker aged The second worker aged The worker aged The first worker aged The second worker aged The third worker aged The worker aged The worker aged

Section Using Simulation to Estimate Probabilities Model: Start at a random place in a table of random digits. Divide the table into pairs of digits. Each pair of digits represents a potential selection. Choose four pairs of digits. Method 1: Assign each worker to a pair of digits from When selecting digits from the table, ignore all pairs other than , and ignore any repeats. Method 2: Since there are 100 two-digit numbers, 100/11 = Assign 9 pairs of digits to each worker: The digits represent worker 1, represent worker 2, etc. When selecting digits from the table, ignore the pair 00, and ignore any digits that represent a worker already selected.

Section Using Simulation to Estimate Probabilities P11a. Researchers at the MacFarlane Burnet Institute for Medical Research and Public Health noticed that the teaspoons had disappeared from their tearoom. They purchased new teaspoons, numbered them, and found that 80% disappeared within 5 months. Suppose that 80% is the correct probability that a teaspoon will disappear within 5 months and that this group purchases ten new teaspoons. Estimate the probability that all the new teaspoons will be gone in 5 months. Start at the beginning of row 34 of Table D on p 828, and add your ten results to the frequency table in Display 5.21, which gives the results of 4990 runs.

Section Using Simulation to Estimate Probabilities P11a. Estimate the probability that all the new teaspoons will be gone in 5 months. Assumptions: You are assuming that each teaspoon has probability 0.80 of disappearing within five months and that whether each spoon disappears is independent of whether other spoons disappear or not. Model: Use single random digits. Assign spoons that disappear (D) the digits 1-8 and spoons that do not disappear (N) the digits 0 and 9. (Notice how this reflects the probability 0.80.) Record the number of spoons that disappear in each run of ten.

Section Using Simulation to Estimate Probabilities P11a. Estimate the probability that all the new teaspoons will be gone in 5 months. Repetition: Starting at row 34 of Table D, the first ten digits are With the assignments given in the Model, this represents: DNDNDNDDND(6 spoons disappeared) This is one run of the simulation.

Section Using Simulation to Estimate Probabilities P11a. Estimate the probability that all the new teaspoons will be gone in 5 months. Repetition:

Section Using Simulation to Estimate Probabilities P11a. Estimate the probability that all the new teaspoons will be gone in 5 months. Conclusion: 517 of the 5000 runs resulted in all ten spoons disappearing, so the estimated probability of all ten spoons disappearing is 517 / 5000, or