LAST EDITED ON 10-30-11 AT 01:21 AM (EST)

LAST EDITED ON 10-29-11 AT 09:59 PM (EST)

Let me try one more time. I'll give you the statistical formula. I'll even give you two different lay-person's explanations for it, to see if that will help. It might not convince you, but I'm hoping it will help anyone out there who's still confused, if such a person exists.

Here's the statistical formula for what we're looking for:
C = n!/{r!(n-r)!}
C is the number of possible combinations we're looking for, n is the total number of objects (in this case 8), and r is the number drawn from those objects (in this case 1). That's 8!/{1!(8-1)!} = 8!/7! = 8 possible combinations, the 8 that I diagrammed in post #40.

(Side note: to show you how this formula works, let's suppose that there were 2 purple rocks in the bag. Then the number of combinations would be 8!/{2!(8-2)!} = 56/2 = 28 combinations. Of those 28 combinations, exactly 7 would assign a purple to any one particular player (I won't draw that out here to save some space, but I can later if you don't see how this works), so the probability of 1 player out of 8 drawing a purple rock when there are 2 purple rocks in a bag of 8 rocks is 7 in 28, or 1 in 4.) Did you follow that?

Here's an explanation that relies less on formulas. I'll call it "Lay-person's explanation #1". Let's assume, for the sake of argument, that the non-purple rocks really do matter. We'll number the rocks from 1-8, with 1 being the purple rock. Anyone who gets rock number 1 will be eliminated. With me so far? The 8 people who draw are labeled from A-H. There are 40,320 possible ways that the 8 rocks can be distributed. (8! = 40,320) And it doesn't matter whether they pick at the same time or one after the other; this is the TOTAL number of possible distributions of the rocks among the 8 players. Here's the beginning and ending of that distributional diagram (of all the possibilities of who might get which rock):

A B C D E F G H
1 2 3 4 5 6 7 8
1 2 3 4 5 6 8 7
1 2 3 4 5 7 6 8
1 2 3 4 5 7 8 6
1 2 3 4 5 8 6 7
1 2 3 4 5 8 7 6
1 2 3 4 6 5 7 8
. . . . . . . .
etc. etc. etc.
. . . . . . . .
8 7 6 5 4 3 1 2
8 7 6 5 4 3 2 1

Obviously I don't have the space to write out all the possibilities here, but you can do it for yourself if you have the time and space. The number of possible permutations (not combinations because order matters, in this example, since we've numbered the rocks) here is 40,320. No more, no less: 40,320. Of those 40,320 permutations, exactly 5,040--no more, no less--have rock #1 going to person A. Exactly 5,040 have rock #1 going to person B. And to person C. And so forth to each person. Oddly enough, 5,040 out of 40,320 reduces to 1 in 8. Go figure. (Ha! I kill me.) In your previous calculation, only 8 out of those 40,320 permutations (1 in 5,040) gives the purple rock to one person, which is obviously not true.

Further, when you add the probabilities, they have to equal 1, because the rock HAS to end up in someone's hand (the probability of someone--anyone, not just a specific person--getting a purple rock is 1 in 1).

Lay person's explanation #2. The probability you are describing would only occur in the following scenario: Say all 8 people drew rocks, one of them being purple (a 1 in 8 chance, as explained above). After that drawing, you eliminate 1 person who didn't get the purple, put the rocks back in the bag after taking a non-purple out, then redraw. Lo and behold, the SAME PERSON got the purple rock! What are the odds, you say? "Well, the probability of that was 1 in 56!" I reply. (1 over 8x7). We eliminate another non-purple person and another rock, put the rocks back in the bag, then redraw. WHOA!!! THE SAME PERSON GOT THE PURPLE ROCK! How freaky! That was a 1 in 336 (1 over 8x7x6) chance! If we keep doing that (eliminating someone who didn't get purple, put the rocks back, then redraw), the probability of the same person getting the purple rock at EVERY DRAWING would be 1 in 5,040.

That's obviously not even close to the scenario they're facing, so that in itself should be a clue that you've been miscalculating.

Now, I wouldn't keep harping on this point if you didn't keep bringing it up, but you also tie it to the point that Cochran was stupidly taking a chance when it was so slight. Well there's a big difference between a 1 in 8 chance (which is what he was facing) and a 1 in 5,040 chance (which is what you've been arguing and much much more unlikely to occur). It still might be a stupid decision, but not at the level that you're arguing.

Edited to take the edge out of my response, because you took out the nastiness in your post, which then made my post look particularly nasty in response. I guess that's how we're playing this game, eh?

Edited further to correct typos and to add a clarification to the explanation.