Originally published in The Reasoner Volume 10, Number 6– June 2016
In Mathematics and Plausible Reasoning: Patterns of plausible inference G. Polya introduces random mass phenomena along the following lines. Consider raindrops falling on an ideally squared pavement, and focus on just two otherwise identical stones called Left and Right. It starts raining (conveniently one drop at a time) and we start recording the sequence of Left and Right according to which stone is hit by each raindrop. In this situation we are (reasonably) unable to predict where the next raindrop will fall, but we can easily predict that in the long run, both stones will be wet. This, Polya suggests, is typical of random mass phenomena: “unpredictable in certain details, predictable in certain numerical proportions to the whole”.
The fact that we can often make reliable predictions on some aggregate, but fail to draw from this obvious conclusions on the individuals, has profound implications not only for the foundations of probability, but also for its practical applications. In medicine, for instance, this is quite the norm. In the absence of further information, what does the fact that a certain side effect of, say statins, is known to affect 1 in 100 patients say about you suffering from it? Problems like this raise the more general question: what is the extent to which forecasts on some aggregate can reliably inform us about its individuals? This question, and its philosophical underpinnings, are tackled by P. Dawid (2016: On Individual Risk, Synthese, First Online, Open Access.)
Here’s a motivating example from the paper