Report

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 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 Maths online AC/on RUN <Exe> OPTN F6 PROB Ran# To Simulate tossing of a coin 1. ◦ Ran# Heads: 0.000 000 -0.499 999 Tails: 0.500 000 – 0.999 999 To simulate LOTTO balls 2. ◦ ◦ 1+40Ran#, truncate the result to 0 d.p., or 0.5+40Ran#, truncate the result to 0 d.p. 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 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 Wednesda 4 y Thursday 5 Friday 6 Saturday 7 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 Calculations Statement of how the calculation needed for the conclusion will be done Number of ‘ successful ’ results Long-run relative frequency = Number of trials Mean = Number of ‘ successful Number of trials ’ results 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: Trial Outcome of Result of trial trial 1 2357 Unsuccessful 2 4635 Successful Number of Trials needed: 30 would be sufficient Calculation: Number of ‘ successful ’ results Probability of 2 boys & 2 girls = Number of trials 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: Trial Toy 1 Toy 2 Toy 3 Toy 4 Toy5 Toy6 1 Y Y Y Y Y Y 10 2 Y Y Y Y Y Y 19 Number of Trials needed: 30 would be sufficient Calculation: Average number of visits = Total visits Number of trials Tally Total Visits 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: Trial Outcome of Trial Result of Trial 1 122 Successful trial 2 167 Unsuccessful trial Number of Trials needed: 30 would be sufficient Calculation: Estimate of probability of ‘passing’ the exam = Number of ‘ successful Number of trials ’ results 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: Trial Outcome of Trial Result of Trial 1 12236754 Successful trial 2 13672987 Unsuccessful trial Number of Trials needed: 30 would be sufficient Calculation: Number Estimate of probability of ‘passing’ the exam = of ‘ successful Number of trials ’ results 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. Tool: 4) Trial: Results: Two numbers (between 1 and 6) will need to be selected first (say 2 & First digit using calculator 1 + 6Ran#, ignore the decimals. 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 Trial Outcome of Trial Result of Trial 1 24 Successful trial 2 13 Unsuccessful trial 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