Have you ever wondered How Gambling Houses make profit?
It involves a little bit of probability. Let's see how it goes.
A Fair Game:
Consider a game in which the outcome can be any number between 0 and 1 i.e. uniform distribution in interval [0,1]. To play each game you need to pay the amount of Rs. 0.5 you get back the amount equal to outcome. (For example if the outcome is 0.6 then you get back Rs. 0.6.).
Let the gambler chooses to play $N$ number of games. In each game he wins or loses $(X_n-0.5)$ amount of rupees where $X_n$ is the outcome of $n$th game. If $N$ is fairly large, the chances are that he neither wins nor loses, meaning that the expected net capital he wins after large number of games is zero.
Let $Y_n=(X_n-0.5)$
Let $S_n=Y_1+Y_2+Y_3+......+Yn$ be the net amount lost or won by the gambler in n plays of the game. In long run the final gain $S_N$ is distributed as Gaussian random variable with zero mean. This results from Strong law of Large Numbers which states that sum of large number of i.i.d.(Independent and Identically Distributed) random variables is Gaussian random variable.
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| Distribution of $S_N$ (zero mean Gaussian) |
Here are some simulation results for fair game for $N=10000$. Observe that in fair game the chances of profit and loss for a Gambling house are equal. In following three instances observe the final value of $S_N$. For first two instances $S_N$ is negative meaning profit for gambling house. But in the third one the $S_N$ is positive meaning that profit to gambler and loss for gambling house.
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| Fig1. Fair Game Instance 1 |
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| Fig.2 Fair Game Instance 2 |
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| Fig3. Fair Game Instance3 |
This is all about the idea of fair game in which neither the gambling house nor gambler can make profit in long run. But Gambling houses are not NGOs. They are there to make profit.
How are they going to make it????
Unfair Game:
Here is the idea --introduce a little unfairness. Make the fees 0.51 instead of 0.5. A little difference. With this minor change the game of chance changes to a large extent in long run. This works because the number of games a Gambler plays is around hundred whereas this number for gambling house is 100000 or 1000000 i.e. the scales of Gambler and Gambling House are different.
Here are the simulation results for short run of the game which is the gambler's scale. (i.e. $N=100$). Observe the value $S_N$ i.e. the value corresponding to rightmost point of the graph
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| Fig.4 Short Run Instance1 |
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| Fig.5 Short run Instance2 |
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| Fig.6 Short Run Instance3 |
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| Fig.7 Short Run Instance4 |
From above Short run instances it's observed that there are still fair chances of gambler to win in short run. In instance 1 and 2 the gambler loses but in the remaining two he wins.
What happens in long run?
Here are simulation results for long run play of the game which is Gambling House's scale i.e $N=100000$
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| Fig.8 Long Run Instance1 |
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| Fig.9 Long Run Instance2 |
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| Fig.10 Long Run Instance3 |
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| Fig.11 Long Run Instance4 |
Four instances for long run of the game are shown here. Notice the difference between these Unfair game Long Run instances(Fig. 1 to Fig.3) and earlier fair game instances(Fig. 8 to Fig.11). Observe that here the net long rub gain of gamblers is always negative meaning that the Gambling house is sure to make profit.
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Ref.
Stochastic Process Limits
by Ward Whitt; AT&T labs (research)
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