If I pulled out a die, red on four sides and green on two, a fair roller in all aspects, and offered to bet you evenly on the outcomes, the proper response is to always bet on red. It doesn't matter if red has come up ten times in a row, and you're thinking "It's got to change soon; maybe I should bet green" - you should always, always, always bet on red with that die.
But after seeing red ten times, a person (you, me, anyone) will almost always have that thought. Because, uh, probability, right?
Actually.... that's wrong.
We have that impression because we want and expect to see a balanced world. We know, by the definition of the test, that in the long run, the die will roll 1 green side for each 2 red sides. And when it doesn't show up soon, we start anticipating it, twitching ahead of it's arrival, because we anticipate the results showing up with the right frequency. But there's no guarantee that this frequency will show itself without a lot of observation - for preference, many copies of the die, rolled many times.
Probabilities require rational construction - you need to think the numbers through, every time.
That said, as clever monkeys, we can cheat the impulse. And it's something we can get better at. Counting cards, for example, is a set of methods for combining problems of frequency and probability in a game where you will see the full set of results given enough time (assuming the whole deck is gone through).
Personally, I think that we should teach card counting in High School. It'd be helpful to students to understand how this stuff works.
Maybe interestingly, I was taught probability in elementary school using playing cards. My father also played poker with me (mercilessly, for any stakes I cared) from the tender age of six or so.
ReplyDeleteMy first thought on seeing, say, ten consecutive heads from a fairly tossed coin, is that it's not a fairly tossed coin.