A simulation imitates a real situation Is supposed to give similar results And so acts as a...

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Experimental Probability and Simulation

Transcript of A simulation imitates a real situation Is supposed to give similar results And so acts as a...

Page 1: A simulation imitates a real situation  Is supposed to give similar results  And so acts as a predictor of what should actually happen  It is a model.

Experimental Probability and Simulation

Page 2: A simulation imitates a real situation  Is supposed to give similar results  And so acts as a predictor of what should actually happen  It is a model.

A simulation imitates a real situation Is supposed to give similar resultsAnd so acts as a predictor of what should

actually happen It is a model in which repeated

experiments are carried out for the purpose of estimating in real life

Simulation

Page 3: A simulation imitates a real situation  Is supposed to give similar results  And so acts as a predictor of what should actually happen  It is a model.

Used to solve problems using experiments when it is difficult to calculate theoretically

Often involves either the calculation of:◦ The long-run relative frequency of an event happening◦ The average number of ‘visits’ taken to a ‘full-set’

Often have to make assumptions about situations being simulated. E.g. there is an equal chance of producing a boy or a girl

Page 5: A simulation imitates a real situation  Is supposed to give similar results  And so acts as a predictor of what should actually happen  It is a model.

AC/on RUN <Exe> OPTN F6 PROB Ran#

Random Numbers on Casio fx-9750G PLUS

Page 6: A simulation imitates a real situation  Is supposed to give similar results  And so acts as a predictor of what should actually happen  It is a model.

1. To Simulate tossing of a coin◦ Ran#

Heads: 0.000 000 -0.499 999 Tails: 0.500 000 – 0.999 999

2. To simulate LOTTO balls◦ 1+40Ran#, truncate the result to 0 d.p., or◦ 0.5+40Ran#, truncate the result to 0 d.p.

Random Numbers (some ideas)

Page 7: A simulation imitates a real situation  Is supposed to give similar results  And so acts as a predictor of what should actually happen  It is a model.

3. To simulate an event which has 14% chance of success

◦ 100Ran#, truncate the result to 0 d.p. 0 – 13 for success, 14-99 for failure, or

◦ 1+100Ran#, truncate the result to 0 d.p. 1-14 for success, 15-100 for failure

Random Numbers

Page 8: A simulation imitates a real situation  Is supposed to give similar results  And so acts as a predictor of what should actually happen  It is a model.

Assume each day has equal probability (1/7)

Use spreadsheet function RANDBETWEEN(1,7)

Generate 4 random numbers to simulate one family

Repeat large number of times

Eg: Simulate probability that 4 members of a family were each born on a different day

Day of the week

Random Number

Sunday 1

Monday 2

Tuesday 3

Wednesday

4

Thursday 5

Friday 6

Saturday 7

Microsoft Office Excel Chart

Page 9: A simulation imitates a real situation  Is supposed to give similar results  And so acts as a predictor of what should actually happen  It is a model.

The description of a simulation should contain at least the following four aspects:

Tools Definition of the probability tool, eg. Ran#, Coin, deck of

cards, spinner Statement of how the tool models the situation

Trials Definition of a trial Definition of a successful outcome of the trial

Results Statement of how the results will be tabulated giving an

example of a successful outcome and an unsuccessful outcome

Statements of how many trials should be carried out

TTRC

Page 10: A simulation imitates a real situation  Is supposed to give similar results  And so acts as a predictor of what should actually happen  It is a model.

TTRC continued

Calculations

Statement of how the calculation needed for the conclusion will be done

Long-run relative frequency =

Mean =

trialsofNumber

results ’successful‘ ofNumber

trialsofNumber

results ’successful‘ ofNumber

Page 11: A simulation imitates a real situation  Is supposed to give similar results  And so acts as a predictor of what should actually happen  It is a model.

Problem: What is the probability that a 4-child family will contain exactly 2 boys and 2 girls?

Page 12: A simulation imitates a real situation  Is supposed to give similar results  And so acts as a predictor of what should actually happen  It is a model.

Tool: First digit using calculator 1+10Ran#Odd Numbers stands for ‘Boy’ andEven Number stands for ‘Girl’

Trial: One trial will consist of generating 4 random numbers to

simulate one family.A Successful trial will have 2 odd and 2 even numbers.

Results:

Number of Trials needed: 30 would be sufficientCalculation:

Probability of 2 boys & 2 girls =

Trial Outcome of trial

Result of trial

1 2357 Unsuccessful

2 4635 Successful

trialsofNumber

results ’successful‘ ofNumber

Page 13: A simulation imitates a real situation  Is supposed to give similar results  And so acts as a predictor of what should actually happen  It is a model.

Problem: As a part of Christmas advertising a petrol station gives away one of 6 Lego toys to each customer who purchases $20 or more of fuel.

Calculate how many visits to the petrol station a customer would need to make on average to collect all 6 Lego toys.

Assumption: The likelihood of one Lego toy being handed out is independent of another.

Page 14: A simulation imitates a real situation  Is supposed to give similar results  And so acts as a predictor of what should actually happen  It is a model.

Tool: Generate random numbers between 1 & 6 (inclusive), each number stands for each toy.

Trial: One trial will consist of generating random numbers till all numbers from 1 to 6 have been generated.Count the number of random numbers need to get one full set

Results:

Number of Trials needed: 30 would be sufficient

Calculation:

Average number of visits = Total visitsNumber of trials

Solution (suggestion)

Trial Toy1

Toy2 Toy3 Toy4 Toy5 Toy6 Tally Total Visits

1 Y Y Y Y Y Y 10

2 Y Y Y Y Y Y 19

Page 15: A simulation imitates a real situation  Is supposed to give similar results  And so acts as a predictor of what should actually happen  It is a model.

Problem: Mary has not studied for her Biology test. She does not know any of the answers on a three-question true-false test, and she decides to guess on all three questions

Design a simulation to estimate the probability that Mary will ‘Pass’ the test. (i.e. guess correct answers to atleast 2 of the 3 questions)

Calculate the theoretical probability that Mary will pass the test.

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Tool: The probability that Mary guesses a question true is one half.First digit using calculator 1 + 10Ran#1to 5 stands for ‘correct answer’6 to 10 stands for ‘incorrect answer’

Trial: One trial will consist of generating 3 random numbers to simulate Mary answering one complete test.

A successful outcome will be getting atleast 2 of the 3 random numbers between 1 and 5.

Results:

Number of Trials needed: 30 would be sufficientCalculation: Estimate of probability of ‘passing’ the exam =

Solution (suggestion)

Trial Outcome of Trial Result of Trial

1 122 Successful trial

2 167 Unsuccessful trial

trialsofNumber

results ’successful‘ ofNumber

Page 17: A simulation imitates a real situation  Is supposed to give similar results  And so acts as a predictor of what should actually happen  It is a model.

Problem: Mary has not studied for her history test. She does not know any of the answers on an eight-question true-false test, and she decides to guess on all eight questions

Design a simulation to estimate the probability that Mary will ‘Pass’ the test. (i.e. guess correct answers to atleast 4 of the eight questions)

Page 18: A simulation imitates a real situation  Is supposed to give similar results  And so acts as a predictor of what should actually happen  It is a model.

Tool: The probability that Mary guesses a question true is one half.

First digit using calculator 1 + 10Ran#

1to 5 stands for ‘correct answer’

6 to 10 stands for ‘incorrect answer’

Trial: One trial will consist of generating 8 random numbers to simulate Mary answering one complete test.

A successful outcome will be getting atleast 4 of the 8 random numbers between 1 and 5.

Results:

Number of Trials needed: 30 would be sufficient

Calculation:

Estimate of probability of ‘passing’ the exam =

Solution (suggestion)

Trial Outcome of Trial Result of Trial

1 12236754 Successful trial

2 13672987 Unsuccessful trial

trialsofNumber

results ’successful‘ ofNumber

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Problem: Lotto 40 balls and to win you must select 6 in any order. In this mini Lotto, there are only 6 balls and you win when you select 2 numbers out of the 6.

Design and run your own simulation to estimate the probability of winning (i.e. selecting 2 numbers out of the 6)Calculate the theoretical probability of winning.

Page 20: A simulation imitates a real situation  Is supposed to give similar results  And so acts as a predictor of what should actually happen  It is a model.

Tool: Two numbers (between 1 and 6) will need to be selected first (say 2 & 4)First digit using calculator 1 + 6Ran#, ignore the decimals.

Trial: One trial will consist of generating 2 random numbersDiscard any repeat numbersA successful outcome will be getting 2 of the 6 random numbers

generatedResults:

Number of Trials needed: 50 would be sufficientCalculation:

Estimate of probability of ‘winning’ = Number of ‘successful’ outcomeNumber of trials

Theoretical probability in this case is 1/15

Solution (suggestion)

Trial Outcome of Trial Result of Trial

1 2 4 Successful trial

2 13 Unsuccessful trial