A TTRC approach to probability simulations

1. Simulations How to design and run a simulation

2. First Step Read your task carefully. What problem exactly are you designing a simulation of? What is it that you need to calculate at the end?

3. Taking a TTRC approach Tool – what tool we’re going to use Trial – how to define a trial, and how many we’ll carry out Results – a table showing our results Calculation – how to use our results to make a calculation which completes the task

4. Tool Remember, your simulation must be based on some probability reasoning. So the Tool the you choose, and how you use it, will have to be based on a probability. E.g. If there’s an even chance of an event happening you could simulate it by tossing a coin, or generating the number 1 or 2 randomly on your calculator.

5. Tool Not quite even chances… If the probability of an event happening is 110 then you’ll probably have to generate a random number from 1 to 10. You could let one number represent the event happening, and all the others represent it not happening. If I generate a random number from 1-10, I could define 1 to mean it happens, and 2-10 mean it doesn’t!

6. How would you simulate… A jogger estimates the probability he’ll finish his run without taking a break is 0.75 What numbers would you generate to simulate whether he takes a break or not? Tool Hmmm.. ?x Ran# + 1

7. You could use… You could use: Random numbers from 1 – 4 where 1,2,3 = no break 4 = takes a break Or Random numbers from 1 – 100 where 1-75 – no break 76-100 = take a break Tool

8. Sample answer: The tool I will use is the random number generator on my calculator. I will generate random numbers from 1 – 4 where: 1, 2, 3 means no break is taken 4 means the jogger takes a break Tool

10. What numbers you’re generating

12. Suppose you have 5 joggers in the group, and the probability that each one might have to take a break is 0.75 How will you simulate this? How many numbers need to be generated per trial? Trial

13. Obviously you need to generate a number for each jogger, so that’s 5 numbers per trial. You’ll need to record these results, and then repeat for 30 trials. Easy! Trial

14. Trial Sample answer: One trial will consist of me generating 5 random numbers as described before, one for each jogger. I will record these numbers, and whether a break was taken or not in a table. I will carry out 30 trials.

16. What you’re going to do with these numbers (duh…record them in a table of course)

18. Results

19. Calculation There are two usual types of calculation made at the end of a simulation. For one of the type discussed here it is usually an estimate of the probability that something will happen.

20. Calculation Estimate the probability the jogging group will not have to take a break. 𝑃𝑛𝑜 𝑏𝑟𝑒𝑎𝑘=# 𝑡𝑟𝑖𝑎𝑙𝑠 𝑤h𝑒𝑟𝑒 𝑛𝑜 𝑏𝑟𝑒𝑎𝑘 𝑡𝑎𝑘𝑒𝑛# 𝑡𝑟𝑖𝑎𝑙𝑠 𝑐𝑎𝑟𝑟𝑖𝑒𝑑 𝑜𝑢𝑡 So the answer will depend on your simulation results.

21. Calculation For a simulation with more open-ended trials the calculation might involve an estimate of average number of times something might happen (e.g. oil strike question practised in class).

22. Life after simulation… You will often be expected to compare your simulation probability to a theoretical one. E.g. Calculate the theoretical probability that the jogging group does not have to take a break, and compare it to your simulation results.

23. The probability that the group doesn’t take a break is really asking for the probability that the first jogger doesn’t need a break and the second doesn’t and the third doesn’t and the fourth doesn’t and the fifth doesn’t. In other words it’s just 0.75 x 0.75 x 0.75 x 0.75 x 0.75