section 5.2 - using simulation to estimate probabilities objectives: 1.learn to design and interpret...

Section 5.2 - Using Simulation to Estimate Probabilities

Objectives:

1. Learn to design and interpret simulations of probabilistic situations


Generating Random Integers

• Using a Table of Random Digits

• Using the TI-83/84 Calculator


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.

Row1 10097 32533 76520 13586 34673 54876 80959 09117 39292 749452 37542 04805 64894 74296 24805 24037 20636 10402 00822 916653 08422 68953 19645 09303 23209 02560 15953 34764 35080 33606

TABLE D Random Digits


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.

Row1 10097 32533 76520 13586 34673 54876 80959 09117 39292 749452 37542 04805 64894 74296 24805 24037 20636 10402 00822 916653 08422 68953 19645 09303 23209 02560 15953 34764 35080 33606

TABLE D Random Digits


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:

71

94

72

13


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.


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.


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.


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.


Model:

Assign each worker a random digit as shown:

Outcome Digit Assigned

The worker aged 27 1



The first worker aged 34 4

The second worker aged 34 5





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.


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.


Model:

Assign each worker a random digit as shown:

Outcome Digit Assigned Set of Digits Assigned

The worker aged 27 01 01-09



The first worker aged 34 04 28-36

The second worker aged 34 05 37-45


The first worker aged 42 07 55-63

The second worker aged 42 08 64-72

The third worker aged 42 09 73-81




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 01 - 11. When selecting digits from the table, ignore all pairs other than 01 - 11, and ignore any repeats.

Method 2:

Since there are 100 two-digit numbers, 100/11 = 9.09. Assign 9 pairs of digits to each worker: The digits 01 - 09 represent worker 1, 10 - 18 represent worker 2, etc. When selecting digits from the table, ignore the pair 00, and ignore any digits that represent a worker already selected.


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.


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.



Repetition:

Starting at row 34 of Table D, the first ten digits are

59808 08391

With the assignments given in the Model, this represents:

DNDNDNDDND (6 spoons disappeared)

This is one run of the simulation.



Repetition:

Run Random Digits Spoons Disappeared1 5980808391 DNDNDNDDND 62 4542726842 DDDDDDDDDD 103 8360949700 DDDNNDNDNN 54 1302124892 DDNDDDDDND 85 7856520106 DDDDDDNDND 86 4605885236 DDNDDDDDDD 97 0139092286 NDDNNNDDDD 68 7728144077 DDDDDDDNDD 99 9391083647 NDNDNDDDDD 7

10 7061742941 DNDDDDDNDD 8



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 0.1034.

Number of Spoons Disappearing Frequency

0 01 02 13 44 245 1336 4607 9738 14969 1392

10 517

Display 5.21

0

200

400

600

800

1000

1200

1400

1600

0 spoons1 spoons2 spoons3 spoons4 spoons5 spoons6 spoons7 spoons8 spoons9 spoons10 spoons

Number of Spoons Disappearing

Frequency

section 5.2 - using simulation to estimate probabilities objectives: 1.learn to design and interpret...

Documents