Sometimes you blog with opinions, other times with questions. This is me blogging with a question - addressed to anyone who's better than I am on theories of probability. Here's the text from which my question will arise:
Joseph Banks Rhine, a psychologist at Duke, had developed an interest in the possibility of extrasensory perception, or E.S.P. Rhine devised an experiment featuring Zener cards, a special deck of twenty-five cards printed with one of five different symbols: a card was drawn from the deck and the subject was asked to guess the symbol. Most of Rhine's subjects guessed about twenty per cent of the cards correctly, as you'd expect, but an undergraduate named Adam Linzmayer averaged nearly fifty per cent during his initial sessions, and pulled off several uncanny streaks, such as guessing nine cards in a row. The odds of this happening by chance are about one in two million. Linzmayer did it three times.
Rhine documented these stunning results in his notebook and prepared several papers for publication. But then, just as he began to believe in the possibility of extrasensory perception, the student lost his spooky talent. Between 1931 and 1933, Linzmayer guessed at the identity of another several thousand cards, but his success rate was now barely above chance. Rhine was forced to conclude that the student's "extra-sensory perception ability has gone through a marked decline."
If something very improbable happens, even something very very very improbable, its happening is not inconsistent with its improbability, right? That the odds of its happening by chance are one in two million is just to say that it's hardly ever going to happen, not that it won't happen at all. Therefore, if it does happen, it could be that this is just that one time in two million. To hypothesize in such circumstances that something might be the case which defies the laws of nature so far as we've got in understanding these seems a big jump. I mean, shouldn't one at least come up with an explanatory hypothesis for E.S.P. first?
To conclude these meanderings with my specific question: how can the mere improbability of an occurrence be taken as evidence for anything when its improbability is perfectly consistent with its actually happening (though not with its happening often)? (Thanks: E.)