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.
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