Howdy Peter, OK, though it's been enjoyable, I won't try any further to convince you. I peppered this message with smileys to let you know that I think mathematical debates are about differing observations, not differing values. In such conversations, its easy to lose sight of that and take something the wrong way. Please don't; it has been fun and just because neither of us has convinced the other (yet) doesn't mean I think the less of you (or, hopefully, the reverse... I know to you, I must seem pretty `thick'). I, myself, would like a little more explanation of _your_ point of view (see my question below beginning with "Why?"). I will recapitulate the high points of my problems with your previous arguments so that you can clear them up for me in private e-mail. I also quote some equations that summarize the point I was trying to make, so that you can examine them and offer up alternatives that represent your point. I am cc'ing cypherpunks on this final message so that they can see these equations. Here we go :-) I wrote a conjecture: SC>A.1 As parishoners play and leave, the division of wealth approaches the SC> `odds' of the game. Which you agreed with: PH>I agree with both conjectures. I then repeated the conjecture in my argument: SC> [A.1] predicts that as ... the number of players goes to infinity, SC> ... the fraction of money won by the church approaches ... the probability SC> the church will win a single trial. Which you do _not_ accept as the statement you agreed with: PH>There is a slight difference between [A.1] and PH>this statement. [A.1] predicts that as ... the number of bets PH>goes to infinity the fraction of bets won will approach ... the PH>probability that the church will win a single trial. On the probability of the player's ultimate ruin you say: PH>Each parishioner has a high probability of losing their savings and a PH>low probability of winning everything owned by the church. It is PH>possible for any single parishioner to win everything, but it is PH>unlikely. Why? Why is the probability not almost `even', like the odds of the game, .51 vs .49? What other information influences this _new_ probability, the probability of the player going broke, if it is not---as I say---the difference in cash resources between the player and the house? I didn't ask you this question in my earlier messages---I thought I was supplying the answer---but you did provide an alternate explanation: PH>This player wins because he or she was fortunate enough to place the PH>first bet in the series [of sufficient consective bets lost by the house]. PH>The player needs to be lucky. To paraphrase my "Why?" question above: can you qualify `lucky'? How `lucky' does the player have to be? I submit to you that given individual trials where the players probability of winning a single unit in a single trial is p, the total amount of money at stake in the series of trials is C, the amount currently held by the player is d, the house C-d=D, that the ultimate chance for the players ruin is given by the equation (from [Weaver] cited in an earlier message): 1-p where r = --- p r^C - r^d R_d (prob. of ruin given d capital) = --------- r^C - 1 Though in the limit (a fair game) you would derive a friendlier form as: d R_d = 1 - - C ...and, of course, at the other extremes, where p=1, or p=0, the player never or always goes broke respectively. These equations are consistent the proposition that the probability of ruin depends on both the odds of the game _and_ the initial distribution of capital. Note their behavior as C increases with respect to d. Soon, this difference dominates even in the face of good `odds'. I invite you to experimentally verify, at your leisure, the `fair game' version with two players and different amounts of pennies where each bet is a single penny and decided by a coin toss. Finally, you offer me this comfort :-) PH>This can be very confusing. I've seen two professional mathematicians PH>and a futures textbook make this mistake. Thank you ;-) If I, two professional mathematicians, a textbook, a book I cited to you, and several other cypherpunks all erred similarly, then it must be a treacherously easy mistake to make; I don't feel any shame. But, I would also relate this little probability parable (again, from [Weaver])---of course drawing no comparisons: In the card room of the Quadrangle Club at the University of Chicago, years ago, a hand con- sisting of thirteen spades was dealt. The celebrated mathematician Leonard Eugene Dickson was one of the players. (Those who know his interest in bridge realize that the probability of his being one of the players was not far below unity.) At the request of his companions, he calculated the probability of this deal (It is roughly 10^-13.) A young know-it-all gaily reported at lunch the next day that he had calculated the probability of dealing thirteen spades, and had found that Dickson had made a mistake. Another famous mathematician, Gilbert Bliss, was present; he properly dressed down the youngster by saying, "Knowing that Dickson calculated a probability and got one result, and you had tried to calculate the same probability but got another result, I would conclude that the probability is practically unity that Dickson was right and you are wrong." Be happy and keep wondering---that's what makes us great, Scott Collins | "That's not fair!" -- Sarah | "You say that so often. I wonder what your basis 408.862.0540 | for comparison is." -- Goblin King ................|.................................................... BUSINESS. fax:974.6094 R254(IL5-2N) collins@newton.apple.com Apple Computer, Inc. 5 Infinite Loop, MS 305-2D Cupertino, CA 95014 ..................................................................... PERSONAL. 408.257.1746 1024:669687 catalyst@netcom.com