martingale casino
July 16th, 2013 at 10:55:41 AM
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This is a paradox I've always wondered about:
What happens if you have a casino (with each game having a house advantage and no table limit) and you have an infinite number of players. Each player has an infinite amount of money, will walk into the casino, bet $1 and keep doubling his bet upon each loss, and walk out once he's won and made the $1 profit. Will this casino lose an infinite amount of money or make an infinite amount of money?
What happens if you have a casino (with each game having a house advantage and no table limit) and you have an infinite number of players. Each player has an infinite amount of money, will walk into the casino, bet $1 and keep doubling his bet upon each loss, and walk out once he's won and made the $1 profit. Will this casino lose an infinite amount of money or make an infinite amount of money?
July 16th, 2013 at 10:59:12 AM
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Is time also infinite? If not, then I think the casino makes money by cutting off some giant losing streaks before they can resolve into 1 unit wins for the players.
"So as the clock ticked and the day passed, opportunity met preparation, and luck happened." - Maurice Clarett
July 16th, 2013 at 10:59:46 AM
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If you start with an infinite amount of money and you win $1, how much do you have after?
"In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice."
-- Girolamo Cardano, 1563
July 16th, 2013 at 11:03:48 AM
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Quote: MathExtremistIf you start with an infinite amount of money and you win $1, how much do you have after?
Infinity and one! Haven't you seen that AT&T commercial? :-)
"So as the clock ticked and the day passed, opportunity met preparation, and luck happened." - Maurice Clarett
July 16th, 2013 at 11:15:40 AM
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Quote: rdw4potusIs time also infinite? If not, then I think the casino makes money by cutting off some giant losing streaks before they can resolve into 1 unit wins for the players.
My gut tells me that whether time is infinite or not, the casino still wins. But I can't quite explain why.
July 16th, 2013 at 11:18:48 AM
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Quote: rdw4potusInfinity and one! Haven't you seen that AT&T commercial? :-)
"I'm sorry, we were looking for infinity plus infinity."
That guy is great!
https://wizardofvegas.com/forum/off-topic/gripes/11182-pet-peeves/120/#post815219
July 16th, 2013 at 11:19:41 AM
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Quote: jonMy gut tells me that whether time is infinite or not, the casino still wins. But I can't quite explain why.
Adding finite and infinite numbers doesn't make sense. That's why the "no table limit" arguments for the effectiveness of the Martingale always break down. There are no casinos nor gamblers with an infinite bankroll, and if you *had* an infinite bankroll, winning a finite amount would be meaningless.
If you had an infinite bankroll, you could make million-dollar blackjack bets for the rest of your life and keep hitting until you busted each hand. You'd lose billions of dollars but you'd still have an infinite bankroll.
Bottom line, in reality both time and money are bounded and finite. In a counterfactual theoretical construct where money is infinite, "winning" and "losing" cease to mean anything.
"In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice."
-- Girolamo Cardano, 1563
July 16th, 2013 at 11:30:25 AM
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deleted
DUHHIIIIIIIII HEARD THAT!
July 16th, 2013 at 11:52:03 AM
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The casino knows: they are the Pushers, the gambling public are the Addicts. Sure some addicts get lucky.... so what?
The winning addict goes to the free buffet and then back to the roulette wheel, he doesn't go home. His wife isn't dressed the way the women in the casino are dressed. The music in the casino is pleasant, the drinks are free ... meanwhile someone just rolled three sevens in a row!
Worry? Why worry? There is the band, there is Helga and tonight who knows. Welcome to Cabaret Casino.
The winning addict goes to the free buffet and then back to the roulette wheel, he doesn't go home. His wife isn't dressed the way the women in the casino are dressed. The music in the casino is pleasant, the drinks are free ... meanwhile someone just rolled three sevens in a row!
Worry? Why worry? There is the band, there is Helga and tonight who knows. Welcome to Cabaret Casino.
July 16th, 2013 at 12:23:40 PM
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If you had more money than the casino...why would you be in the casino for anything other than entertainment?
"I am your average American gambling idiot" - Me
July 16th, 2013 at 12:56:54 PM
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I'll tell you why: Interest.Quote: jonMy gut tells me that whether time is infinite or not, the casino still wins. But I can't quite explain why.
In this imaginary universe, the casino has no debt and can actually reinvest their money. So they earn interest until the day you finally win.
I invented a few casino games. Info:
http://www.DaveMillerGaming.com/ —————————————————————————————————————
Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
July 16th, 2013 at 8:19:36 PM
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Eventually someone realizes if he stops the infiniteness after making an infinite offer to buy the casino, he will win it all and collects all the money.
I am a robot.
July 16th, 2013 at 8:44:05 PM
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Quote: jonThis is a paradox I've always wondered about:
What happens if you have a casino (with each game having a house advantage and no table limit) and you have an infinite number of players. Each player has an infinite amount of money, will walk into the casino, bet $1 and keep doubling his bet upon each loss, and walk out once he's won and made the $1 profit. Will this casino lose an infinite amount of money or make an infinite amount of money?
If I have infinite amount of money, I will bet $$$$$$ not $1 and Make the $$$$$$$ profit or bet infinite amount of money and Make infinite amount of money.
Stephen Au-Yeung (Legend of New Table Games®) NewTableGames.com
July 16th, 2013 at 9:45:14 PM
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Quote: MrCasinoGamesIf I have infinite amount of money, I will bet $$$$$$ not $1 and Make the $$$$$$$ profit or bet infinite amount of money and Make infinite amount of money.
If you have an infinite amount of money, why would you bet $$$$ not $1? You already have an infinite amount of money and you could lose it by betting an infinite amount of money! Talk about greedy!
Anyway, if Eliot J, Wizard, Crystal Math or any of the other math heads here can actually answer my original question with a mathematical solution/proof, I'd appreciate it.
July 16th, 2013 at 10:07:30 PM
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Quote: jonIf you have an infinite amount of money, why would you bet $$$$ not $1? You already have an infinite amount of money and you could lose it by betting an infinite amount of money! Talk about greedy!
Anyway, if Eliot J, Wizard, Crystal Math or any of the other math heads here can actually answer my original question with a mathematical solution/proof, I'd appreciate it.
Jon,
I would like to know the actually answer to your original question with a mathematical solution/proof too.
P.S. Good question.
Stephen Au-Yeung (Legend of New Table Games®) NewTableGames.com
July 16th, 2013 at 10:26:38 PM
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Did you know that infinitely comes in different sizes?
July 17th, 2013 at 12:05:15 AM
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Quote: jonIf you have an infinite amount of money, why would you bet $$$$ not $1? You already have an infinite amount of money and you could lose it by betting an infinite amount of money! Talk about greedy!
Anyway, if Eliot J, Wizard, Crystal Math or any of the other math heads here can actually answer my original question with a mathematical solution/proof, I'd appreciate it.
Okay: I'm assuming you mean an even-money wager with a house edge, so you've got W probability of winning, 0 < W < 0.5, and 1-W = L probability of losing. If that's so, then for every bet level 2^N from N=0..+inf, more players will lose than win. Regardless of how many players end up winning and quitting, at any given point in time the casino is always winning more from the bettors than losing to them, so the casino wins. Consider |P| is the size of the set of players. It's infinite, but that turns out not to matter. After the $1 bet level, W * |P| of the bettors quit but the casino has won (L-W) * |P|. The losers from the previous round are L*|P|. They each bet $2. Of those, W*L*|P| win and quit; the other L*L*|P| lose. But the casino still wins more on that round because L > W. It keeps going: on round T, each player bets 2^T. There are L^T players still betting, but the casino still wins more of those players' bets than it loses: (2L)^T*|P|*(L-W). That's a positive number.
In short, at no point in time does the casino ever take a loss, so their aggregate result must be a profit.
"In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice."
-- Girolamo Cardano, 1563
July 17th, 2013 at 12:10:52 AM
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Loving the Cabaret references; was my first show in undergrad. "Wilkommen, bienvenue, welcome!"
Also get a kick out of the AT&T guy; he's good with everybody from kids to b-ballers.
Also get a kick out of the AT&T guy; he's good with everybody from kids to b-ballers.
If the House lost every hand, they wouldn't deal the game.
July 17th, 2013 at 2:49:51 AM
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Quote: jonThis is a paradox I've always wondered about:
What happens if you have a casino (with each game having a house advantage and no table limit) and you have an infinite number of players. Each player has an infinite amount of money, will walk into the casino, bet $1 and keep doubling his bet upon each loss, and walk out once he's won and made the $1 profit. Will this casino lose an infinite amount of money or make an infinite amount of money?
I misread it to mean the casino also had an infinite amount of money, but if they don't each player will win his $1 eventually and because the number of players are also infinite, the casino will lose an infinite amount of money. Martingdale can't lose with an infinite BR and no bet limits.
I am a robot.
July 17th, 2013 at 3:04:50 AM
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Quote: onenickelmiracleI misread it to mean the casino also had an infinite amount of money, but if they don't each player will win his $1 eventually and because the number of players are also infinite, the casino will lose an infinite amount of money. Martingdale can't lose with an infinite BR and no bet limits.
I disagree. If the casino starts with a finite bankroll, I think they will end up with an infinite bankroll (or grow their bankroll to an amount which is infinitesimally smaller than an infinite bankroll). Here is my reasoning why:
There are infinite players. This implies that every possible thing that can happen, will happen, and infinitely many times. Therefore every player must fall into one of the following two categories:
1) Those who make their $1 and quit
2) Those who never win and keep doubling their bets infinitely
It is the very fact that there are infinite players that allows group 2 to exist. And since there are infinite players, there are infinite players in each group.
Because each player has an infinite bankroll, no player's bankroll can ever be exhausted, no matter how much they lose.
Therefore:
1) The players in group 1 will continue to have their infinite bankrolls.
2) The players in group 2 are effectively sharing their infinite bankrolls with the casino, but this does not decrease any group 2 player's bankroll because it is infinite.
Therefore the casino's bankroll becomes infinite (or approaches an infinite bankroll) as a result of the players in group 2.
July 17th, 2013 at 6:56:06 AM
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I really can't resist this:
"You can't run out of time. Time is infinite! You are finite. Zathras is finite. This... this is wrong tool." Zathras in Babylon 5's "War Without End" (But of coruse, no one ever listens to poor Zathras).
"You can't run out of time. Time is infinite! You are finite. Zathras is finite. This... this is wrong tool." Zathras in Babylon 5's "War Without End" (But of coruse, no one ever listens to poor Zathras).
Donald Trump is a fucking criminal
July 17th, 2013 at 3:03:37 PM
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The players taking their money out winning $1 each causing the casino to pay out a final transaction for a net loss for an infinite amount of players. The losers never really lose since their losses are only theoretical and they are never stopped from playing.
We add a guaranteed infinite loss to a never guaranteed infinite win and I have trouble believing the casino doesn't lose an infinite amount of money.
We add a guaranteed infinite loss to a never guaranteed infinite win and I have trouble believing the casino doesn't lose an infinite amount of money.
I am a robot.
July 17th, 2013 at 5:29:32 PM
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First round, the casino will win an infinite number of bets and lose an infinite number of bets. So they will have to pay out $1 to the winners an infinite number of times, and take in $1 from the losers and infinite number of times. The casino can structure these payout such that they end up with any amount of profit or loss, or such that they make an infinite profit, or such that they make an infinite loss.
For example, to make an infinite profit, the casino collect losers 1&2, pays winner 1. They repeat this process for losers 3&4 and winner 2, etc. In sequence, they collect losers 2k-1 & 2k, paying winner k for k=1,2,3,4.... The casino will end up with a $1 after each k, so end up with an infinite profit.
If instead the casino pays winner 2k-1 & k, and collects loser k for k=1,2,3.... they will end up paying out $1 for each k, and so end up with infinite loss.
But say the casino only wants to only win $10, then collect losers 1-10, then pay winner k and collect loser k+10 for k=1,2,3..... For each k, the casino ends up even, so the casino makes $10.
This is a feature of rearranging the order of an infinite series that does not converge absolutely (that is, the sum of absolute values does not converge): You can rearrange them to approach any value.
Interestingly, it does not matter to the casino if the game is in their favor or not, the players could even have an advantage. The casino only needs to be in charge of what order they pay out and collect.
Actually, the game could be a 99.99% winner for the players with a payout of a billion-to-one, and the casino could still make an infinite amount of profit.
For example, to make an infinite profit, the casino collect losers 1&2, pays winner 1. They repeat this process for losers 3&4 and winner 2, etc. In sequence, they collect losers 2k-1 & 2k, paying winner k for k=1,2,3,4.... The casino will end up with a $1 after each k, so end up with an infinite profit.
If instead the casino pays winner 2k-1 & k, and collects loser k for k=1,2,3.... they will end up paying out $1 for each k, and so end up with infinite loss.
But say the casino only wants to only win $10, then collect losers 1-10, then pay winner k and collect loser k+10 for k=1,2,3..... For each k, the casino ends up even, so the casino makes $10.
This is a feature of rearranging the order of an infinite series that does not converge absolutely (that is, the sum of absolute values does not converge): You can rearrange them to approach any value.
Interestingly, it does not matter to the casino if the game is in their favor or not, the players could even have an advantage. The casino only needs to be in charge of what order they pay out and collect.
Actually, the game could be a 99.99% winner for the players with a payout of a billion-to-one, and the casino could still make an infinite amount of profit.
July 17th, 2013 at 6:29:20 PM
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Practically a player with infinte money would not need to play at a casino, because he can spend as much as he wants and never run out of money. Think, he can buy the world a zillion times over and he'd still have an infinite amount of money. Unless he also has infinte time (as in he'll live for eternity), then he's engaging in a colossal waste trying to win $1, or $1,000,000,000,000,000,000,000,000 from the casino. It makes no difference to his bottom line.
Donald Trump is a fucking criminal
July 17th, 2013 at 8:10:10 PM
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Quote: NareedPractically a player with infinte money would not need to play at a casino, because he can spend as much as he wants and never run out of money. Think, he can buy the world a zillion times over and he'd still have an infinite amount of money. Unless he also has infinte time (as in he'll live for eternity), then he's engaging in a colossal waste trying to win $1, or $1,000,000,000,000,000,000,000,000 from the casino. It makes no difference to his bottom line.
Would someone who has an infinite amount of money and an infinite lifespan run out of money?
July 17th, 2013 at 8:21:33 PM
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Quote: jonWould someone who has an infinite amount of money and an infinite lifespan run out of money?
Not if things have finite cost. But that wouldn't matter after a few millennia anyway because we'd solve resource scarcity and then money would be meaningless.
"In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice."
-- Girolamo Cardano, 1563
July 17th, 2013 at 8:29:13 PM
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This thread makes me want an infinite drink!
https://wizardofvegas.com/forum/off-topic/gripes/11182-pet-peeves/120/#post815219
July 17th, 2013 at 8:50:39 PM
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Quote: Mission146This thread makes me want an infinite drink!
Jon is probably that trucker you turned away for years until he finally gave in-should have given him your best deal.
I am a robot.
July 17th, 2013 at 10:09:36 PM
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Quote: jonWould someone who has an infinite amount of money and an infinite lifespan run out of money?
All bets are of at the End of Eternity.
Other than that, no :P
Donald Trump is a fucking criminal
July 19th, 2013 at 1:53:00 AM
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A game
that is designed to earn particular % must do what it was designed to do if given
enough bets... So as more closer to infinity - as more accurate the %....
that is designed to earn particular % must do what it was designed to do if given
enough bets... So as more closer to infinity - as more accurate the %....
July 31st, 2013 at 4:41:20 PM
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Quote: jonThis is a paradox I've always wondered about:
What happens if you have a casino (with each game having a house advantage and no table limit) and you have an infinite number of players. Each player has an infinite amount of money, will walk into the casino, bet $1 and keep doubling his bet upon each loss, and walk out once he's won and made the $1 profit. Will this casino lose an infinite amount of money or make an infinite amount of money?
You would need to define when a casino with unlimited funds has lost an infinite amount of money.
What you can do is ask: if the casino has funds F, at what probability P(F) will it lose all of its funds. Once you have P(F), you can look at P(F) for F -> infinity. This would be some definition to your original question.
Keeping that last step F -> infinity in mind, lets assume the casino has limited funds F. Let's also assume the casino offers a "fair" game which is actually winnable by the player even at a remote chance p > 0.
Now here is my central statement: Any single player will make 1 unit profit with probability 1.0.
The player - by use of the martingale system - will only need a single win to make 1 unit profit. The probability p of such a win is greater than zero (by the definition of the "fair" game). By the law of large numbers to win W bets his winning ratio W/N for N bets played will tend to p with probability of 1.0 for N -> infinity.
Since N -> infininity also W -> infinity. Especially W >= 1. Since the player can afford an infinite number of bets, he will win at least a single bet with probability of 1.0. WIth the martingale system, once he has won a single bet, he won the unit (and quits).
Now each single player will eventually make $1 profit with probability 1.0. Thus for a single unit funded casino to get broke, it is P(1) = 1.0.
By the same argument (each player will win 1 unit with probability 1.0), the probability of a casino with funds F will get his funds reduced to F-1 by a single player is also 1.0. So the probability of getting broke with funds F can be shifted to the probability of getting broke with funds F-1 by P(F) = 1.0 * P(F-1).
Solving the recursing P(F) = 1.0^(F-1) * P(1) = 1.0 (for any positive F).
Hence, Following my special definition of an "infinitely funded casino", taking the limit F -> infinity still makes P(F) -> 1.0, and thus the MangoJ-infinitely funded casino will get broke almost surely.
August 1st, 2013 at 12:04:50 PM
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Mango, I don't think that's correct. After thinking about this some more, and reading some of the very insightful posts above, I have to conclude that the casino would always win an infinite amount of money.
"Any single player will make 1 unit profit with probability 1.0." - I think this is the fallacy in your argument. Any single player will make 1 unit profit with a probability approaching 1 (but not equal to 1). If time goes out to infinity, there will always be some very unlucky players that are still playing (and losing).
Consider the games for all players are played simultaneously each second (e.g, first game played for $1 at second 0, second game for $2 played at second 2, etc.) , and the game is such that the player has a 51% chance of losing and a 49% chance of winning even money. if P is the number of players, after the first second, the casino wins $1*(.51p-.49p) as p->infinity, which is an infinite number. After the first second, there are still .51 * the original number of players (which is still an infinite number of players) playing (at a bet of $2), so the casino wins $2*(.02p*.51) as p-> infinity, which is an infinite number. For any given second t, the casino wins ($2^t*.02*p*.51^t) as p-> inf, which is still an infinite number. Note that the number of remaining players at time t is .51^t which would always be infinite since there is an infinite number of starting players. The total amount the casino wins is thus: SUM (1->t->inf) ($2^t*.02*p*.51^t) as p-> inf which is infinite.
In fact, this casino can actually start out in huge debt and yet make an infinite amount of money.
I guess the reason why the casino wins is because you can't "get" to infinity
"Any single player will make 1 unit profit with probability 1.0." - I think this is the fallacy in your argument. Any single player will make 1 unit profit with a probability approaching 1 (but not equal to 1). If time goes out to infinity, there will always be some very unlucky players that are still playing (and losing).
Consider the games for all players are played simultaneously each second (e.g, first game played for $1 at second 0, second game for $2 played at second 2, etc.) , and the game is such that the player has a 51% chance of losing and a 49% chance of winning even money. if P is the number of players, after the first second, the casino wins $1*(.51p-.49p) as p->infinity, which is an infinite number. After the first second, there are still .51 * the original number of players (which is still an infinite number of players) playing (at a bet of $2), so the casino wins $2*(.02p*.51) as p-> infinity, which is an infinite number. For any given second t, the casino wins ($2^t*.02*p*.51^t) as p-> inf, which is still an infinite number. Note that the number of remaining players at time t is .51^t which would always be infinite since there is an infinite number of starting players. The total amount the casino wins is thus: SUM (1->t->inf) ($2^t*.02*p*.51^t) as p-> inf which is infinite.
In fact, this casino can actually start out in huge debt and yet make an infinite amount of money.
I guess the reason why the casino wins is because you can't "get" to infinity
August 1st, 2013 at 5:50:37 PM
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I think OPs premise has one too many infintes. Let's change the premise a bit:
Everything as posted, but the casino can only accommodate 1000 players for up to 1000 spins (thinking of roulette) a day.
In this scenario, the casino will go broke despite a 5.3% (or even 50.3%) house edge.
I was gonna post the proof, but the statistical numbers are too ridiculously small. For all intents and purposes, let's just say 1000 losses in a row is never gonna happen while mankind exists. So the casino loses 1000 units a day everyday.
Martingale works provided there are no table limits, and you have a finite but unlimited bankroll.
Everything as posted, but the casino can only accommodate 1000 players for up to 1000 spins (thinking of roulette) a day.
In this scenario, the casino will go broke despite a 5.3% (or even 50.3%) house edge.
I was gonna post the proof, but the statistical numbers are too ridiculously small. For all intents and purposes, let's just say 1000 losses in a row is never gonna happen while mankind exists. So the casino loses 1000 units a day everyday.
Martingale works provided there are no table limits, and you have a finite but unlimited bankroll.
August 27th, 2013 at 7:58:41 AM
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Quote: MangoJNow here is my central statement: Any single player will make 1 unit profit with probability 1.0.
1/2 + 1/4 + 1/8 + 1/16 + 1/32... does equal 1.0 http://en.wikipedia.org/wiki/1/2_%2B_1/4_%2B_1/8_%2B_1/16_%2B_⋯
But do we have to consider that blackjack, craps, roulette and baccarat are not even money games?
Let's just say we play a casino game that has a 47% chance the player wins
47/100 + 235/1000 + 1175/10000 + 5875/100000... converges at something less than 1.0
It feels like I'm going down the wrong path, but just something that came into my mind and leads me to believe the probability that any one player wins $1 is less than 1.0
August 27th, 2013 at 10:30:06 AM
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Quote: TomGLet's just say we play a casino game that has a 47% chance the player wins
47/100 + 235/1000 + 1175/10000 + 5875/100000...
Not sure where you get these numbers.
If the game has a chance p > 0 that the player will win, the sequence should be
p + (1-p) * p + (1-p)^2 * p + ...
= p * sum (1-p)^n from n=0 to infinity
= p * 1 / (1 - (1-p)) (geometric series)
= 1
August 27th, 2013 at 10:42:04 AM
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I think you're right, but still not sure if this is the right path. At any given point in time, there will be an infinite number of winners and an infinite number of losers. So why does that change after an infinite number of balckjack hands?
August 27th, 2013 at 3:37:48 PM
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Assuming all variables are infinite, players, seats, money, wins, losses, free drinks, people paying for drinks, the casino will always be ahead.
Examples:
100 players cash out for a 100
100 players lost 4 hand in a row each losing 15 bucks.
100 to players
1500 to casino.
Net casino 1400
The players cash out 1 at a time and there will always be 100 players seated.
As one player leaves a new place takes the open seat
All things being equal everyone is a winner.
Examples:
100 players cash out for a 100
100 players lost 4 hand in a row each losing 15 bucks.
100 to players
1500 to casino.
Net casino 1400
The players cash out 1 at a time and there will always be 100 players seated.
As one player leaves a new place takes the open seat
All things being equal everyone is a winner.
Buffering...
August 30th, 2013 at 2:14:03 PM
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Quote: MangoJNot sure where you get these numbers.
If the game has a chance p > 0 that the player will win, the sequence should be
p + (1-p) * p + (1-p)^2 * p + ...
= p * sum (1-p)^n from n=0 to infinity
= p * 1 / (1 - (1-p)) (geometric series)
= 1
I don't think you can say it equals 1 (you can't get to infinity), I think it is proper to say it approaches 1. Which is why the graph of the casino's profits will always go up as time goes on to infinity. For any time from 1 to infinity, you will always have an infinite number of "unlucky" players playing.
September 9th, 2013 at 2:45:18 PM
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Here's a very interesting documentary on infinity.
It's from the BBC Horizon series, a well respected science program.
It takes a while to get going and if you don't have an interest in this sort of thing you may find it too dry to sit through.
Best viewed in 720 HD
http://www.youtube.com/watch?v=BvWq3pxPeRo
It's from the BBC Horizon series, a well respected science program.
It takes a while to get going and if you don't have an interest in this sort of thing you may find it too dry to sit through.
Best viewed in 720 HD
http://www.youtube.com/watch?v=BvWq3pxPeRo
A photon without any luggage checks into a hotel, he's travelling light.
September 9th, 2013 at 5:52:53 PM
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My problem with the casino making a profit in this scenario is the losses are never solid because the losses must be held to pay eventual wins and player wins of $1 are banked. A casino can't make a profit unless it can take money out of the game, but in this case the money can never be known to be a profit.
I am a robot.
