The Paradox of Hot and Cold Streaks

“People think random distributions should be uniform.” – Keith Law

Despite his name, Eubulides was not a Teletubby. He was, in fact, a philosopher from Ancient Greece. Let’s consider his Bald Man Paradox:

A man with a full head of hair is obviously not bald. Now the removal of a single hair will not turn a non-bald man into a bald one. And yet it is obvious that a continuation of that process must eventually result in baldness.

This is known better as the Paradox of the Heap, which replaces the hair with grains of sand. If the grains of sand are removed, one by one, at which point is the sand no longer a heap of sand?

This same issue applies to players said to be in the midst of hot or cold streaks. At what point is a player’s performance so good that he is hot? If he had one fewer hit in his last 15 at-bats, would he still be hot? What about two fewer?