Experimental Probability and Simulation

1. Experimental Probability and Simulation

2. 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

3. 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

4. Simulating tossing a fair coin • Maths online

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

6. 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.

7. 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

8. 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

9. 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

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

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

12. 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 =

13. 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.

14. 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

15. 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 questionsDesign 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.

16. 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 =

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

18. 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 =

19. 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.

20. 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