Experimental Probability and Simulation

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

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

Simulating tossing a fair coin Maths online

Random Numbers on Casio fx-9750G PLUS AC/on RUN <Exe> OPTN F6 PROB Ran#

Random Numbers (some ideas) To Simulate tossing of a coin Ran# Heads: 0.000 000 -0.499 999 Tails: 0.500 000 – 0.999 999 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 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

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

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

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

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

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

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.

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

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.

Solution (suggestion) 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 sufficient Calculation: Estimate of probability of ‘passing’ the exam = Trial Outcome of Trial Result of Trial 1 122 Successful trial 2 167 Unsuccessful trial

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)

Solution (suggestion) 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 = Trial Outcome of Trial Result of Trial 1 12236754 Successful trial 2 13672987 Unsuccessful trial

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 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.

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