Sunday, 19 June 2011

The first theoretical probability analysis was for one die. The second was for two. Both are distinct monuments in the history of ideas.

Cardano is given credit with being the first to correctly calculate a theoretical probability.  I disagree.  FN David, whose book seems to be a key reference in many subsequent books on the history of probability, gives Cardano this credit while being aware that for many centuries before, empirical calculation of the likelihoods of dice outcomes (and outcomes from other kinds of randomisation machines) were known.

Specifically, the people who took the given astragalus and chipped and rendered it into a die must have been working towards a theoretical goal of equi-probability.  They must get credit.  They kept chipping away until they reached their goal - a tolerable or measurable closeness to a theoretical model of equi-probability.  This may seem to trivial to be a calculation, but it surely was.  And it was even perhaps literally a calculation too, since the word has its origin in the latin work for a small counting pebble made of the easily manipulable limestone.

What Cardano did was correctly showed how you really needed to enumerate all possible combinations of complex outcomes, to see through equivalence classes to complete enumerations.  Someone else, lost forever to history, introduced the idea that the fraction of elementary outcomes over the universe of elementary outcomes was a significant fraction.  We had up until then been lulled into making the wrong choices between permutations and combinations due to the similarity of two (or more) dice.  If humanity had started rolling identically crafted randomisation machines with clear and distinct markings on them, perhaps we would have got it earlier.  If the pair of dice were marked with 1 to 6 on the first and 7 to 12 on the second then it might have clicked earlier.  But it didn't.  And Cardano (or his side-kick Ferrari) get the credit for disambiguating combinations and permutations.  I'll do another posting on combinations and permutations, which are always badly explained in all probability books I've read.  It is still, I think, quite a non-intuitive idea for us to understand.

What Cardano's work has enabled, however, is the possibility of the correct analysis of the repeat of the same experiment many times.The rolling of two dice, of one die any number of times, etc.  By showing us how to properly discover the elementary sample space, he pioneered a significant piece of the statistical approach to scientific experimental design and has quickened the pace and reliability of science itself.

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