Fudbucker,
- I agree with your idea, but would present it with a different situation.
Say that you find a deck of cards in the closet and decide to play some solitaire or something.
You sit down at the table and turn over the first card. It's an ace of spades. You place the ace back in the deck, shuffle the cards and once again, turn over the first card. This time, it's the ace of diamonds. Hmm. So, you try the same thing again. This time, you get the ace of spades again.
'Wait a minute…' You do it one more time, and this time, you get the ace of hearts.
If you’re paying attention, you’re growing suspicious about this deck you found in the closet. You’re starting to suspect that you don’t have the ordinary deck that you had assumed. But, why is that? Why are you suspicious?
You’re suspicious because the probability of drawing that 'hand' is so small if the deck is a normal deck.
Let’s try that again with a new deck. But, this time, the first card you draw is a 3 of diamonds, the second is a
Jack of spades, the third is a 9 of clubs and the fourth is a 9 of hearts. In this case, you probably are not suspicious.
But, of course you realize that the probability of drawing that hand, given a normal deck, is just as small as the probability of drawing that previous hand…
So, what’s the problem here? Why are you not suspicious of this deck, when you were suspicious of the first one?
It turns out that there are two factors causing you to be suspicious of that first deck -- and one is missing in regard to the second deck. There is nothing about the second hand that sets it apart in such a way as to suggest another plausible hypothesis… If there were, you’d be suspicious of that second deck as well. It’s as simple as that…
If, for instance, you were to remember that a few years back you had assembled a deck of 3’s, jacks, and 9’s (for some weird reason), the second hand would also stand out and would also make you suspicious. And, again, your suspicion would be mathematically justified…
Any time that new information you receive causes the overall information you have to logically favor a new ‘worldview,’ you are justified in revising your expectations accordingly. Who could argue with that?
And, that’s what happened when you drew the 4 aces, but did not happen when you drew the other hand.
The nature of a random "sample" has mathematical (probabalistic) implications about the
population the sample was taken from. The 4 aces, by being so improbable under your preexisting worldview regarding the deck, and by being so probable under a different, plausible, worldview about the deck, caused the overall information you had to logically favor the new worldview.
The "nondescript" hand, by not being especially probable under any other plausible worldview about the deck (that we can think of), even though it was specifically so improbable under the old worldview, does not cause the overall information you have to logically favor a new worldview.
By not suggesting another worldview, and thereby not setting itself apart from most other
possible hands, the nondescript hand "blends" with most other possible hands and effectively takes on the combined probability of most hands…