SimulationSimulation

SimulationSimulation

SimulationSimulation imitation of chance imitation of chance behavior based on a model that behavior based on a model that accurately reflects the phenomenon accurately reflects the phenomenon under considerationunder consideration

By observing simulated outcomes, By observing simulated outcomes, researchers gain insight on the real researchers gain insight on the real world.world.

SimulationSimulation

Why use simulation?Why use simulation?

Some situations do not lend themselves to Some situations do not lend themselves to precise mathematical treatment. precise mathematical treatment.

Others may be difficult, time-consuming, Others may be difficult, time-consuming, or expensive to analyze. In these or expensive to analyze. In these situations, simulation may approximate situations, simulation may approximate real-world results; yet, require less time, real-world results; yet, require less time, effort, and/or money than other effort, and/or money than other approaches.approaches.

How to Conduct a How to Conduct a SimulationSimulation

A simulation is useful only if it closely mirrors A simulation is useful only if it closely mirrors real-world outcomes. The steps required to real-world outcomes. The steps required to produce a useful simulation are below:produce a useful simulation are below:

1.1. State the problem or describe the random State the problem or describe the random phenomenon.phenomenon.

2.2. State the assumptions.State the assumptions.3.3. Assign digits to represent outcomes Assign digits to represent outcomes (die, spinner, random (die, spinner, random

digit table, calc.)digit table, calc.)

4.4. Simulate many repetitions Simulate many repetitions (until outcome shows stable pattern)(until outcome shows stable pattern)

5.5. State your conclusions.State your conclusions.

Simulation ExampleSimulation Example

In this section, we work through an example In this section, we work through an example to show how to apply simulation methods to to show how to apply simulation methods to probability problems.probability problems.

Problem DescriptionProblem Description

On average, suppose a baseball player hits a On average, suppose a baseball player hits a home run once in every 10 times at bat. home run once in every 10 times at bat. Using simulation, estimate the likelihood that Using simulation, estimate the likelihood that the player will hit 2 home runs in consecutive the player will hit 2 home runs in consecutive at bats.at bats.

SolutionSolution

Use five steps required to produce a usefulUse five steps required to produce a usefulSimulation:Simulation:1.1. Problem: Problem:

– Estimate the likelihood that the player will hit 2 home runs in Estimate the likelihood that the player will hit 2 home runs in consecutive at batsconsecutive at bats

2.2. State Assumptions: State Assumptions: – at bats are independent of each other (that is, what happens during at bats are independent of each other (that is, what happens during

one at bat does not influence the next at bat)one at bat does not influence the next at bat)– Homerun occurs 10% of at batsHomerun occurs 10% of at bats

3.3. Assign Digits: Assign Digits: – two possible outcomes: home run or no home runtwo possible outcomes: home run or no home run– ““2” represents a home run and other digits represent no home run2” represents a home run and other digits represent no home run– ““22” represents back to back home runs 22” represents back to back home runs – Any other two digit combination represents failure to hit back to back Any other two digit combination represents failure to hit back to back

homerunshomeruns

4.4. Simulate Many Repetitions:Simulate Many Repetitions:– Stat Trek’s random digit generator was utilized for this exampleStat Trek’s random digit generator was utilized for this example– In this example, the list of random numbers consists of 500 2-digit In this example, the list of random numbers consists of 500 2-digit

pairspairs

5.5. State ConclusionsState Conclusions– In the list, we found 6 occurrences of "22", which are highlighted in red In the list, we found 6 occurrences of "22", which are highlighted in red

in the tablein the table

Random NumbersRandom Numbers

42 99 02 65 04 14 30 09 70 88 89 85 95 40 53 67 25 50 48 79 86 42 99 02 65 04 14 30 09 70 88 89 85 95 40 53 67 25 50 48 79 86 92 76 24 53 39 08 73 78 17 72 81 08 01 68 94 43 43 95 12 36 90 92 76 24 53 39 08 73 78 17 72 81 08 01 68 94 43 43 95 12 36 90 28 88 34 69 18 69 91 79 14 82 26 94 15 26 19 41 74 02 17 20 38 28 88 34 69 18 69 91 79 14 82 26 94 15 26 19 41 74 02 17 20 38 84 74 30 34 96 09 46 61 41 02 93 94 90 00 71 84 98 30 82 80 11 84 74 30 34 96 09 46 61 41 02 93 94 90 00 71 84 98 30 82 80 11 92 97 81 29 85 44 40 05 83 92 97 81 29 85 44 40 05 83 2222 04 86 13 33 00 99 74 75 27 43 68 04 86 13 33 00 99 74 75 27 43 68 2222 59 20 66 00 24 01 96 84 19 14 57 26 47 58 51 73 06 08 49 52 59 20 66 00 24 01 96 84 19 14 57 26 47 58 51 73 06 08 49 52 70 15 79 35 65 28 40 77 93 73 33 24 25 70 15 79 35 65 28 40 77 93 73 33 24 25 2222 32 03 89 03 62 13 85 32 03 89 03 62 13 85 16 23 28 12 61 16 75 45 37 15 54 36 18 45 64 31 31 06 80 32 75 16 23 28 12 61 16 75 45 37 15 54 36 18 45 64 31 31 06 80 32 75 99 27 91 25 98 05 55 32 27 16 51 45 89 31 78 90 82 05 11 39 80 99 27 91 25 98 05 55 32 27 16 51 45 89 31 78 90 82 05 11 39 80 83 01 20 10 67 97 33 72 09 98 78 39 56 57 54 63 35 21 35 93 18 83 01 20 10 67 97 33 72 09 98 78 39 56 57 54 63 35 21 35 93 18 17 48 55 60 44 92 21 07 77 42 46 86 41 49 76 96 36 62 38 11 64 17 48 55 60 44 92 21 07 77 42 46 86 41 49 76 96 36 62 38 11 64 07 04 58 23 56 29 37 87 37 59 47 83 77 21 63 10 95 87 10 42 71 07 04 58 23 56 29 37 87 37 59 47 83 77 21 63 10 95 87 10 42 71 12 88 06 52 42 99 02 65 04 14 30 09 70 88 89 85 95 40 53 67 25 12 88 06 52 42 99 02 65 04 14 30 09 70 88 89 85 95 40 53 67 25 50 48 79 86 92 76 24 53 39 08 73 78 17 72 81 08 01 68 94 43 43 50 48 79 86 92 76 24 53 39 08 73 78 17 72 81 08 01 68 94 43 43 95 12 36 90 28 88 34 69 18 69 91 79 14 82 26 94 15 26 19 41 74 95 12 36 90 28 88 34 69 18 69 91 79 14 82 26 94 15 26 19 41 74 02 17 20 38 84 74 30 34 96 09 46 61 41 02 93 94 90 00 71 84 98 02 17 20 38 84 74 30 34 96 09 46 61 41 02 93 94 90 00 71 84 98 30 82 80 11 92 97 81 29 85 44 40 05 83 30 82 80 11 92 97 81 29 85 44 40 05 83 2222 04 86 13 33 00 99 74 04 86 13 33 00 99 74 75 27 43 68 75 27 43 68 2222 59 20 66 00 24 01 96 84 19 14 57 26 47 58 51 73 59 20 66 00 24 01 96 84 19 14 57 26 47 58 51 73 06 08 49 52 70 15 79 35 65 28 40 77 93 73 33 24 25 06 08 49 52 70 15 79 35 65 28 40 77 93 73 33 24 25 2222 32 03 89 32 03 89 03 62 13 85 16 23 28 12 61 16 75 45 37 15 54 36 18 45 64 31 31 03 62 13 85 16 23 28 12 61 16 75 45 37 15 54 36 18 45 64 31 31 06 80 32 75 99 27 91 25 98 05 55 32 27 16 51 45 89 31 78 90 82 06 80 32 75 99 27 91 25 98 05 55 32 27 16 51 45 89 31 78 90 82 05 11 39 80 83 01 20 10 67 97 33 72 09 98 78 39 56 57 54 63 35 05 11 39 80 83 01 20 10 67 97 33 72 09 98 78 39 56 57 54 63 35 21 35 93 18 17 48 55 60 44 92 21 07 77 42 46 86 41 49 76 96 36 21 35 93 18 17 48 55 60 44 92 21 07 77 42 46 86 41 49 76 96 36 62 38 11 64 07 04 58 23 56 29 37 87 37 59 47 83 77 62 38 11 64 07 04 58 23 56 29 37 87 37 59 47 83 77

This simulation predicts that the player will hit This simulation predicts that the player will hit consecutive home runs 6 times in 500 at bats. consecutive home runs 6 times in 500 at bats.

Thus, the simulation suggests that there is a 1.2% Thus, the simulation suggests that there is a 1.2% chance that a randomly selected pair of at bats chance that a randomly selected pair of at bats would consist of two home runs. would consist of two home runs.

The actual probability, based on the The actual probability, based on the multiplication rule, states that there is a 1.0% , states that there is a 1.0% chance of hitting consecutive home runs. chance of hitting consecutive home runs.

While the simulation is not exact, it is very close. While the simulation is not exact, it is very close. And, if we had generated a list with more random And, if we had generated a list with more random numbers, it likely would have been even closer.numbers, it likely would have been even closer.

Orders of frozen yogurt flavors have the following Orders of frozen yogurt flavors have the following relative frequencies: 38% chocolate, 42% vanilla, relative frequencies: 38% chocolate, 42% vanilla, 20% strawberry.20% strawberry.

1.1. State problem.State problem.– Simulate 10 frozen yogurt sales based on this recent Simulate 10 frozen yogurt sales based on this recent

history.history.

2.2. State assumptions.State assumptions.– Frequencies are the same as statedFrequencies are the same as stated– Assume that customers order one flavor onlyAssume that customers order one flavor only– Customers’ choices of flavors do not influence one Customers’ choices of flavors do not influence one

anotheranother

3.3. Assign digits.Assign digits.– 00-37 chocolate00-37 chocolate– 38-79 vanilla38-79 vanilla– 80-99 strawberry80-99 strawberry

4.4. Simulate.Simulate.5.5. State your conclusions.State your conclusions.

Simulate: Use Calculator to simulateSimulate: Use Calculator to simulate– Results: Run 20 numbersResults: Run 20 numbers

State Conclusions:State Conclusions:– Chocolate:Chocolate:– Vanilla:Vanilla:– Strawberry:Strawberry:

A couple plans to have children until they have a girl or until A couple plans to have children until they have a girl or until

they have four children, whichever comes firstthey have four children, whichever comes first..

Problem: Problem: – Simulate and estimate probability that the strategy will Simulate and estimate probability that the strategy will

produce girlproduce girl

Assumptions:Assumptions:– Probability of obtaining girl .5 and boy .5Probability of obtaining girl .5 and boy .5– Sexes of successive children independentSexes of successive children independent

Assign Digits (or flip coin):Assign Digits (or flip coin):– 0, 1, 2, 3, 4 = girl0, 1, 2, 3, 4 = girl– 5, 6, 7, 8, 9 = boy5, 6, 7, 8, 9 = boy

Simulate using line 130 of table B:Simulate using line 130 of table B:

Conclusions:Conclusions:

HomeworkHomework

6.1-4, 8-9, 12