Computer simulations of random experiments

The ideas and techniques discussed in this reading set the conceptual foundation for the interpretation of probabilities used in statistical models to make inference based on data.

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1. Introduction

“We have no right to take anything for granted,” Holmes answered. “It is certainly ten to one that they go down stream, but we cannot be certain.”

“Winwood Reade is good upon the subject,” said Holmes. “He remarks that, while the individual man is an insoluble puzzle, in the aggregate he becomes a mathematical certainty. You can, for example, never foretell what any one man will do, but you can say with precision what an average number will be up to. Individuals vary, but the percentages remain constant. So says the statistician.”

Arthur Conan Doyle (author) quotation from the Sherlock Holmes book The Sign of Four (1913).

Random experiments pinpoint and illuminate the concept of probability, and how using them may help to explore and even ‘measure’ the uncertainty of various outcomes of interest.

Such experiments involve a large number of observations or ‘trials’, where the outcome of each single observation is uncertain (random) and, therefore, unpredictable.

However, summary statistics for large datasets, such as the sample mean, become surprisingly non-random with the growth of the sample size.

In particular, the relative frequencies of a specific outcome in a long series of trials get closer and closer to some idealised ‘limit’, which can be interpreted as the probability of that outcome.

It may be tedious or difficult to perform such experiments in real terms. Still, we can leverage the power of computers by deploying their ‘random number generators’ (RNG), which can programmatically emulate randomness.

This approach is often termed Monte Carlo simulations (referring to the Monte Carlo casino in Monaco).