Experimental Probability and Simulation Simulation A simulation imitates

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

� 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-9750 G PLUS � AC/on � RUN <Exe> � OPTN � F 6 � PROB � Ran#

Random Numbers (some ideas) To Simulate tossing of a coin 1. ◦ Ran# � Heads: 0. 000 -0. 499 999 � Tails: 0. 500 000 – 0. 999 To simulate LOTTO balls 2. ◦ ◦ 1+40 Ran#, truncate the result to 0 d. p. , or 0. 5+40 Ran#, truncate the result to 0 d. p.

Random Numbers 3. To simulate an event which has 14% chance of success ◦ 100 Ran#, truncate the result to 0 d. p. � 0 – 13 for success, 14 -99 for failure, or ◦ 1+100 Ran#, 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) Day of the � Use spreadsheet function week RANDBETWEEN(1, 7) Sunday � Generate 4 random numbers to Monday Tuesday simulate one family Wednesda � Repeat large number of times y Random Number 1 2 3 4 Thursday 5 Friday 6 Saturday 7

TTRC 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 =

The Structure of your Simulation assessment 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 � Justify your choice. Results: � Carrying out the simulation and recording outcomes � Statement of how the results will be tabulated giving an example of a successful outcome and an unsuccessful outcome Calculations: � Selecting and using appropriate measures. � Write a conclusion which uses your calculations to justify your choice.

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

Tool: First digit using calculator 1+10 Ran# 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 1 2357 Unsuccessful 2 4635 Successful Number of Trials needed: 30 would be sufficient Calculation: Probability of 2 boys & 2 girls =

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: Trial Toy 1 Toy 2 Toy 3 Toy 4 Toy 5 Toy 6 1 Y Y Y 10 2 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

Problem: Mary has not studied for her Biology test. She does not know any of the answers on a threequestion 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 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 + 10 Ran# 1 to 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 =

Problem: Mary has not studied for her history test. She does not know any of the answers on an eightquestion 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 + 10 Ran# 1 to 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: Estimate of probability of ‘passing’ the exam =

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 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 + 6 Ran#, 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: 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