Wednesday, 20 March 2013

Probability preferences: the irrelevance of parallel/sequential distinction

In a sense, whether you throw one die sequentially n times to get a $6^n$ event space, or whether you simultaneously toss n distinguishable dice at one time, it doesn't matter.  As long as you read your die results in a way which preserves the identity of the die the number appears.  I'll leave off talking about what implication this has for the famous Pascal-Fermat problem of points until a later posting.  For now, consider what this means for the classic repeated experiment in probability theory.  If the events are genuinely independent, then it doesn't matter what relative time it is when you toss each one.  The law of large numbers could equally well be satisfied with a single massively parallel experiment in, say, tossing a coin than it is in tossing a coin sequentially n times.

Likewise in set theory, there's a curious atemporality to Venn diagrams.   And when discussing the joint probability of $A \cap B$, which is of course not the same as A then B.  Even with Bayes' theorem it is important to realise that the 'given' meaning in A|B is with respect to our knowledge of the occurrence of B, not that B happened first and then A subsequently happened.