Designing and interpreting a simulation
A valid simulation preserves the target category proportions and states what happens to the device between trials. A completed run gives a relative frequency, which is an estimate and can vary from the theoretical probability
- Match category proportions Map equal device outcomes so each target category has its required probability \(\displaystyle \frac{2}{8}=\frac{1}{4}\)
- Count one selection as one trial The repeated unit must be one complete selection, not the whole run \(\displaystyle 1\text{ selection}=1\text{ trial}\)
- Reset for independent trials With replacement, restore the initial state before the next trial \(\displaystyle \text{reset before each trial}\)
- Interpret the estimate Divide successful trials by total trials and allow finite-run variation \(\displaystyle \widehat{p}=\frac{\text{successful trials}}{\text{total trials}}\)
Worked example
A target experiment gives A a probability of three-tenths and B a probability of seven-tenths. A simulation uses 10 equal outcomes, with 3 mapped to A and 7 mapped to B, and resets before each trial. One run records 8 A outcomes in 30 trials. Find the relative frequency for A
- \(\displaystyle \frac{3}{10}=\Pr(A),\quad \frac{7}{10}=\Pr(B)\)The device mapping has the same category proportions as the target experiment
- \(\displaystyle 1\text{ selection}=1\text{ trial},\quad \text{reset before each trial}\)The repeated unit and reset rule preserve independent trials
- \(\displaystyle s=8,\quad t=30\)The run has 8 successful A trials out of 30 trials in total
- \(\displaystyle \widehat{p}_A=\frac{8}{30}=\frac{4}{15}\)The exact relative frequency is the success count divided by the total trial count
Common mistakes
Difficulty levels
Questions about this topic
How does a device mapping reproduce a target probability?+
When all device outcomes are equally likely, assign a matching proportion of those outcomes to each target category. For example, two of eight outcomes gives a probability of one-quarter
Why must a with-replacement device reset before each trial?+
Resetting restores the same outcome proportions, so the trials remain independent with the same probabilities
Why can a simulation estimate differ from the theoretical probability?+
A finite run is only one set of results. Its relative frequency can be above or below the theoretical probability, while longer runs often give closer estimates
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 14: Statistical inference