Pretend the casino is run out of a church. "Parishioners" arrive and enter a confessional to place their bets. The "priest" cannot see who is placing each bet. Each "parishioner" plays until he or she is broke. "Parishioners" arrive at a steady rate and will do so indefinitely.
Let me just make sure I understand what you mean. I believe you are saying: Conjecture A: A.1 As parishoners play and leave, the division of wealth approaches the `odds' of the game. Thus if the odds are .51 house (of God), .49 parishoner, then eventually the house will end up with 51 cents out of every dollar `played'. Just as it would if the church were playing against one very wealthy parishoner (i.e., the `world'). A.2 Since there are a large number of parishoners, enough games can always be played to make the distribution match the odds. If this is _not_ what you mean to say then I apologize for missing your point; read no further---just send me explanations to clear up my mis-understanding. If Conjecture A is accurate statement of your belief, then please step across this line. ---------- Let me walk through your model, one parishoner at a time. Please read this with an open mind; it could be true.
Each "parishioner" plays until he or she is broke.
Lets say the odds of the game are .51 to .49. Each parishoner has $100. Each parishoner plays until broke. At some point in play, the distribution of wealth with respect to _that player_ may be arbitrarily close to c=$51, p=$49. What, though, is the distribution at the _end_ of that game? Since each game only ends when the p=$0, the distribution is c=$100, p=$0. On to the next parishoner. After the 9th, but before the 10th parishoner, the distribution must be c=$900, p[10]=$100. It can't be worse than that for the church, or we wouldn't have moved on to the 10th parishoner. It can't be better for the player because each has only $100 to wager. After the n'th, c=$100n, p[n+1]=$100. Conjecture A predicts that as n, the number of players, goes to infinity, c, the fraction of money won by the church, approaches C, the probability the church will win a single trial. But in fact, the model shows that as n approaches infinity, c goes to 1. Where could one disagree with this interpretation of the model? a. Maybe the church has 10 confessionals, or 1000, or 10,000. Serializing the players might be a `paper' advantage to the church that doesn't occur in reality. b. Players can have any amount of money, not just $100 dollars. c. What if the church goes broke? (a) Imagine that the church has at most k confessionals, and thus can play no more than k simultaneous games. Fill all k. All other players are waiting in line for an open spot. The next parishoner can't play until an existing player goes broke. The distribution of wealth during play by the (k-1+10)th player is exactly as before, except now it is +/-$100(k-1). (b) has no impact. As above, at the end of each game the fraction of money won by the church with respect to that player is 1 (assuming it's the player and not the church that `went out'). (c) If the church goes broke, all bets are off, literally but not figuratively. The distribution of wealth is c=0, P=1 (P for all players as opposed to p for a single player). This also does not match the expectation of .51.
The chance of the "church" to win or lose is the same on every bet, regardless of who places it.
That is true. But the only way the player can realize his mathematical expectations is if he is allowed to continue playing even after he is out of money (i.e., so he can climb back out of the hole). Ok, the first player goes out, but the infinity of players after him can make up for that, right? Wrong, because on his way to winning back the first players money, if the second player goes broke, _his_ game is over. Now its up the third guy, ad infinitum (literally)..... just because the series is infinite doesn't mean the sum is. No set of players, all of whom go broke, break the church. Therefore, for the series to end it must be instigated by a set of players that includes at least one who doesn't go broke (i.e., the church goes broke instead). In fact, a single player who doesn't go broke ends the series without any help from other players. Thus, to stem the tide of pious donations (i.e., the church's winnings), a single player with enough money to `outlast' the church is required. Hope you found this interesting but not insulting, 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