Probability 7 - Fair gambling game

Seventh post in the series.

Fair gambling game

Fair gambling game is obviously a game that when you ask someone what his earnings will be before he starts to play he will tell you that it will be zero on average (otherwise the game system gives him some advantage over the “bank”, since if on average he is better off, this means that its not the luck that he may relay on but the gambling system prefers him over the “bank”)

Now this is tricky question, it will be crucial for the interpretation of rigorous mathematic approach to come - If you ask a gambler, a priori (you ask hin on time \(n=0\) before he even played his first bet), what will be, on average, his gain be on the \(n+1\) step (the surplus on this single bet) after he already have played \(n\) games? He will tell you it is still zero. And indeed, what difference it makes a priori to know that there will be moment in future where he has played \(n\) games? If it does make the difference Otherwise it is build into the system that before playing gambler already knows that after playing \(n\) games he on average wins in the next turn, making the whole game not fair.

In last post on probability I talked about filtration I tried to explain how filtration is related to information gathering as time progresses. I used the ability to answer questions regarding the outcome of events to illustrate the relation. Now the question previously asked can be stated as follows (combining the conditional expectation, that relay on appropriate filtration).

Suppose the gain per unit gamble on time \(n\) is given by random variable \(X_n\), Now our previous question rigorously translates to

• First question: \(\mathbb{E}[X_n]=0\)
• Second, the tricky one: \(\mathbb{E}[X_{n+1}|\mathcal{F_n}]=0 (a.s.)\)