Track 4 · Probability · Lesson 9 of 9
Theory versus simulation
Estimate probabilities by simulation, explain the law of large numbers, and judge how far an observed frequency can reasonably be from the true probability.
A risk analyst can't work out by hand the probability that a portfolio loses more than 10% in a year, so she simulates 10 000 possible years on a computer and finds such a loss in 700 of them. Is the probability 0.07 to within ±0.005, or only to within ±0.05? Simulation is one of the most practical tools in probability — as long as you know how much to trust its output.
There are two ways to arrive at a probability.
- Theoretical probability comes from a model: a fair die has 6 equally likely faces, so .
- Empirical probability (an observed relative frequency) comes from repeating the experiment: if 11 of 60 rolls show a six, the empirical probability is .
The two rarely agree exactly. The empirical value is an estimate of the theoretical one, and it varies from one set of trials to the next. A simulation is just a cheap way of generating many trials with a computer instead of real dice, customers or markets.
Included with a Formula Quest membership
It has 8 graded exercises with feedback, hints and worked solutions. After it you will be able to:
- Distinguish theoretical probabilities from empirical (observed or simulated) frequencies
- Estimate a probability by simulation and give its standard error
- Work out how many simulated trials a required precision needs
- Explain the law of large numbers as dilution, and recognise the gambler's fallacy