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. 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

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:
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.

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, for failure, or
1+100Ran#, truncate the result to 0 d.p.
1-14 for success, 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
Day of the week
Random Number
Sunday 1
Monday 2
Tuesday 3
Wednesday 4
Thursday 5
Friday 6
Saturday 7

9. 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

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 =
Trial
Outcome of trial
Result of trial 1
2357
Unsuccessful 2
4635
Successful

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
Trial
Toy1
Toy2
Toy3
Toy4
Toy5
Toy6
Tally
Total Visits 1 Y 10 2 19

15. 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.

16. 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

17. 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)

18. 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

19. 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.

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
Trial
Outcome of Trial
Result of Trial 1
2 4
Successful trial 2 13
Unsuccessful trial