Independent Events and Chance :
Each roll of the dice on the casino floor, every time you want to throw snake eyes in Vegas, you are hit by a little thing in probability called : Independent Events.
Independent Events refers to the fact that every time you attempt to gamble in a game of chance [i.e. chance based games as opposed to poker etc.] your chances with every attempt remain the same - the probabilistic return on a coin toss is 1/2 for a fair coin, 1/37 of 1/38 for a game of roulette and so on and so forth. Hence the word "hot-streak" is a misnomer.
Games which have no memory are the ones which cannot be beaten without an element or elements of unfair play. That's why black-jack is so different, and that's why to an extent "counting cards" in black-jack actually works - because its a memory based game.
Engineers with background in DSP : Digital Signal processing would know that signals processing is a memory based system : i.e there is an element of causality because of the interference of a signal with a channel during transmission. This memory is required to be extracted to nullify the effects of the channel : i.e. for inter-symbol interference and noise reduction.
But games of chance are just that - once a dice rolls, it rolls without caring about the past - its only statistically after rolling the dice a few hundred thousand or a million times, we can determine the actual distribution of each event and the standard deviation for the same.
Even after this, if you play craps, you have as good a chance to win on the next roll. There is still the element of "house advantage" that causes the dreaded Gambler ruin. The house advantage is not simply based on statistics of winning alone. That is, its not just the odds against the gambler at the casino. The main factor is the payout factor.
Take the two-column roulette example. Bet $13 per column, $26 total. A win nets you $12. But the odds were 26-to-12, so a bet with no house advantage would pay $12. The casino's profit comes by grabbing the $26 when you lose but forking over a lousy $12 when you win.
Casinos make money in the following ways:
1. EDGE : It's the fraction of the overall amount bet that the casino would earn if every set of decisions fell precisely into statistical line. So theoretically, they could do worse or better in a short period of time, but remember, this is a statistical average, not an intstantaneous prop!
Consider the two-column roulette bet again. In 38 spins, the house expects to win 12 rounds at $26 each for $312 in all, and to lose 26 rounds at $10 each for a total of $260. The total bet would be $26 x 38 or $988 while theoretical "take" is $312 - $260 or $52. The edge is $52 divided by $988 or 5.26 percent. A particular series of 38 rounds may not give the casino 24 wins and 14 losses. But, over 1000's of such throws and the house can haul their 5.26 percent to the bank in humvee-limos :-) : Do the math!
2. Return % : It is the very opposite of EDGE, and relates to the amount that the owner is going to get back on his investement in a % term of course. Here to you could weigh in how much you would make over 10's or 100's of pulls of the slot machine lever. This is for the casinos to figure out the money thats going to be drained out of their enormous pockets.
US Government stipulation ensures that the machines that people muck around with actually have a chance of giving a return on their hard earned quarters. But that's a pittance compared to what the casino stiffs you for.
Sunday, August 23, 2009
Beating Las Vegas : The Gambler's ruin & Winning Strategy
People head out to Vegas with often elaborate schemes to beat the house at a game of black-jack or roulette. Other card games like poker involve a certain degree of understanding of the game's nuances and are not a pure guessing game.
The Gambler's ruin is a true and hard fact buttressed by probabilistic evidence.
Let two players each have a finite number of coins (say, n_1 for player one and n_2 for player two). Now, flip one of the coins (from either player), with each player having 50% probability of winning, and transfer a coin from the loser to the winner. Now repeat the process until one player has all the pennies.
If the process is repeated indefinitely, the probability that one of the two player will eventually lose all his pennies must be 100%. In fact, the chances P_1 and P_2 that players one and two, respectively, will be rendered bankrupt are
P_1 = (n_2)/(n_1+n_2)
(1)
P_2 = (n_1)/(n_1+n_2),
(2)
i.e., your chances of going bankrupt are equal to the ratio of coins your opponent starts out to the total number of pennies.
Therefore, the player starting out with the smallest number of coins has the greatest chance of going bankrupt. Even with equal odds, the longer you gamble, the greater the chance that the player starting out with the most coins wins. Since casinos have more coins than the gamblers, this principle allows casinos to always come out ahead in the long run. And the common practice of playing games with odds skewed in favor of the house makes this outcome just that much quicker.
Its an interesting argument. Your lucky streak is a terminal short-run phase. The longer you stay in a winning streak the less likely you are to cash out. Hence the moment you hit a jackpot, make for the cash counters, because its not going to last.
The analysis above however states that there are games which are "skewed in favor of the house". If they aren't of course, the casino doesn't make much money now does it - but the question is how does any advantage truly convert itself into "house advantage" - as with two players who have no prior knowledge a priori on the true outcome of every event, and with the expectation that the game itself is player in a fair and unbiased manner how could the house hold the upper hand in a game?
The only way then for a Gambler to beat the house without cheating is to
1. Have some sort of advantage in the game, with the game having an inbuilt bias towards the gambler
2. Have more money bankrolled than the casino and expect that in a truly unbiased and fair contest the casino gets bankrupt in the longest run. This of course does not consider the fact that casinos can close the table if they are losing a lot of money, to protect their own interests, since the gambler too is given a chance to walk away with his profit/loss.
In games where the possible set of sample points are large - even if the payout is unimaginably large, the playing field truly screws the gambler for all his money. This is because his chance of hitting it are speciously low, while the payout for the house is a zero sum game. In 38 holes in roulette for example, its possible that the house pays out 34 to 35 times the original amount, while is still a little less than what the house would make whenever it wins - and it would wild 37/38 times.
Very simply put - if you are consistent in your choice each time - your chance of winning is 1 - (37/38)^n - the longer you stay in the better your chances, and depending on how much money you stake, and when the roulette wheel averages out, your payout will cause you that much ruin.
If the house sets a minimum and maximum bet limit, statistically it can never lose money. For the gambler the odds are always 1/38. For house its always 1/38. This is because each spin is a new spin.
One interesting betting system is the Martingale betting system :
The simplest of these strategies was designed for a game in which the gambler wins his stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double his bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake. Since a gambler with infinite wealth will with probability 1 eventually flip heads, the Martingale betting strategy was seen as a sure thing by those who practised it. Of course, none of these practitioners in fact possessed infinite wealth, and the exponential growth of the bets would eventually bankrupt those who choose to use the Martingale.
Hence if you bet x to win >= x, this strategy doesn't seem bad, if the probability of you winning atleast x is a good shot - i.e. you are not unduly biased in losing with respect to the house. Else the Martingale, in my opinion, just doesn't work, when you don't have much of a shot. Since your betting amount doubles each time, it doesn't take too long until you go bankrupt - since your losses mount like crazy. Also the payout amount isn't anything special, you only end up with x at the end of your Martingale run.
Beating the house seems a plausible option iff the game gives the house no more than a 50% chance at winning - and the gambler has a decent enough bankroll, and house doesn't set too high of a limit to wager a bet. The payout is obviously proportional to the amount dropped in, but the odds act solely a multiplier of the initial wager. Winning against the house is never a foolproof strategy, and that's why you have so many people at the slot machines in casinos in Vegas. You might as well lose your $ in quarters than risk it all at the blackjack dealer.
The Gambler's ruin is a true and hard fact buttressed by probabilistic evidence.
Let two players each have a finite number of coins (say, n_1 for player one and n_2 for player two). Now, flip one of the coins (from either player), with each player having 50% probability of winning, and transfer a coin from the loser to the winner. Now repeat the process until one player has all the pennies.
If the process is repeated indefinitely, the probability that one of the two player will eventually lose all his pennies must be 100%. In fact, the chances P_1 and P_2 that players one and two, respectively, will be rendered bankrupt are
P_1 = (n_2)/(n_1+n_2)
(1)
P_2 = (n_1)/(n_1+n_2),
(2)
i.e., your chances of going bankrupt are equal to the ratio of coins your opponent starts out to the total number of pennies.
Therefore, the player starting out with the smallest number of coins has the greatest chance of going bankrupt. Even with equal odds, the longer you gamble, the greater the chance that the player starting out with the most coins wins. Since casinos have more coins than the gamblers, this principle allows casinos to always come out ahead in the long run. And the common practice of playing games with odds skewed in favor of the house makes this outcome just that much quicker.
Its an interesting argument. Your lucky streak is a terminal short-run phase. The longer you stay in a winning streak the less likely you are to cash out. Hence the moment you hit a jackpot, make for the cash counters, because its not going to last.
The analysis above however states that there are games which are "skewed in favor of the house". If they aren't of course, the casino doesn't make much money now does it - but the question is how does any advantage truly convert itself into "house advantage" - as with two players who have no prior knowledge a priori on the true outcome of every event, and with the expectation that the game itself is player in a fair and unbiased manner how could the house hold the upper hand in a game?
The only way then for a Gambler to beat the house without cheating is to
1. Have some sort of advantage in the game, with the game having an inbuilt bias towards the gambler
2. Have more money bankrolled than the casino and expect that in a truly unbiased and fair contest the casino gets bankrupt in the longest run. This of course does not consider the fact that casinos can close the table if they are losing a lot of money, to protect their own interests, since the gambler too is given a chance to walk away with his profit/loss.
In games where the possible set of sample points are large - even if the payout is unimaginably large, the playing field truly screws the gambler for all his money. This is because his chance of hitting it are speciously low, while the payout for the house is a zero sum game. In 38 holes in roulette for example, its possible that the house pays out 34 to 35 times the original amount, while is still a little less than what the house would make whenever it wins - and it would wild 37/38 times.
Very simply put - if you are consistent in your choice each time - your chance of winning is 1 - (37/38)^n - the longer you stay in the better your chances, and depending on how much money you stake, and when the roulette wheel averages out, your payout will cause you that much ruin.
If the house sets a minimum and maximum bet limit, statistically it can never lose money. For the gambler the odds are always 1/38. For house its always 1/38. This is because each spin is a new spin.
One interesting betting system is the Martingale betting system :
The simplest of these strategies was designed for a game in which the gambler wins his stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double his bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake. Since a gambler with infinite wealth will with probability 1 eventually flip heads, the Martingale betting strategy was seen as a sure thing by those who practised it. Of course, none of these practitioners in fact possessed infinite wealth, and the exponential growth of the bets would eventually bankrupt those who choose to use the Martingale.
Hence if you bet x to win >= x, this strategy doesn't seem bad, if the probability of you winning atleast x is a good shot - i.e. you are not unduly biased in losing with respect to the house. Else the Martingale, in my opinion, just doesn't work, when you don't have much of a shot. Since your betting amount doubles each time, it doesn't take too long until you go bankrupt - since your losses mount like crazy. Also the payout amount isn't anything special, you only end up with x at the end of your Martingale run.
Beating the house seems a plausible option iff the game gives the house no more than a 50% chance at winning - and the gambler has a decent enough bankroll, and house doesn't set too high of a limit to wager a bet. The payout is obviously proportional to the amount dropped in, but the odds act solely a multiplier of the initial wager. Winning against the house is never a foolproof strategy, and that's why you have so many people at the slot machines in casinos in Vegas. You might as well lose your $ in quarters than risk it all at the blackjack dealer.
Labels:
casino,
counting cards,
gambler,
gambler ruin,
game theory,
martingale,
math,
mathetmatics,
mgm grand,
probability,
set theory,
statistics,
vegas
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