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**Experimental Probability and Simulation**

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

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**Often involves either the calculation of:**

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

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**Simulating tossing a fair coin**

Maths online

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**Random Numbers on Casio fx-9750G PLUS**

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

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**Random Numbers (some ideas)**

To Simulate tossing of a coin Ran# Heads: Tails: – To simulate LOTTO balls 1+40Ran#, truncate the result to 0 d.p., or 0.5+40Ran#, truncate the result to 0 d.p.

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

100Ran#, truncate the result to 0 d.p. 0 – 13 for success, for failure, or 1+100Ran#, truncate the result to 0 d.p. 1-14 for success, for failure

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

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 Day of the week Random Number Sunday 1 Monday 2 Tuesday 3 Wednesday 4 Thursday 5 Friday 6 Saturday 7

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**TTRC Tools Trials Results**

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

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**TTRC continued Calculations**

Statement of how the calculation needed for the conclusion will be done Long-run relative frequency = Mean =

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

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Tool: First digit using calculator 1+10Ran# Odd Numbers stands for ‘Boy’ and Even 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 sufficient Calculation: Probability of 2 boys & 2 girls = Trial Outcome of trial Result of trial 1 2357 Unsuccessful 2 4635 Successful

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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.

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**Solution (suggestion)**

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 visits Number of trials Trial Toy1 Toy2 Toy3 Toy4 Toy5 Toy6 Tally Total Visits 1 Y 10 2 19

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**Problem: Mary has not studied for her Biology test**

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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**Solution (suggestion)**

Tool: The probability that Mary guesses a question true is one half. First digit using calculator Ran# 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 sufficient Calculation: Estimate of probability of ‘passing’ the exam = Trial Outcome of Trial Result of Trial 1 122 Successful trial 2 167 Unsuccessful trial

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**Problem: Mary has not studied for her history test**

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)

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**Solution (suggestion)**

Tool: The probability that Mary guesses a question true is one half. First digit using calculator Ran# 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 = Trial Outcome of Trial Result of Trial 1 Successful trial 2 Unsuccessful trial

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**Problem: Lotto 40 balls and to win you must select 6 in any order**

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 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.

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**Solution (suggestion)**

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 numbers Discard any repeat numbers A successful outcome will be getting 2 of the 6 random numbers generated Results: Number of Trials needed: 50 would be sufficient Calculation: Estimate of probability of ‘winning’ = Number of ‘successful’ outcome Number of trials Theoretical probability in this case is 1/15 Trial Outcome of Trial Result of Trial 1 2 4 Successful trial 2 13 Unsuccessful trial

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