Question about Baccarat (hand played correlation to Probability of a Casino Loss)

Hello Wizz! Thank you for brilliant site and forum.
My question is regarding correlation between hands played and Casino Loss probability. This theme described in "Casino Operations Management" - a book written by Jim Kilby & Jim Fox . I've found an example table in this book which shows correlation between number of hands played and Casino loss percentage (this table (see below) generated for 100000$ flat betting a Banker sessions). I can understand perfectly well how the Standard Deviation and the Mean or Expected Value (Theo. win) has been calculated...Unfortunately I can't find a formula or way how to calculate "Probability of casino loss" represented at this table.
Is this somehow related to +2.33 Z -2.33 Z ?
...Or it should be calculated using Central limit theorem? Please help to understand.

EDIT: Actually, no, I think it was the Central Limit Theorem. I just had rounding error.

The probability of a Casino loss is P(X<0), where X is the result after N bets. A reasonable(*) probability distribution is a Gaussian of mean N*m0 and standard deviation sqrt(N)*s0, where m and s0 is the mean and standard deviation of a single bet.

(*) This probabiltiy distribution is correct in the central limit N->infinity, and hence is "reasonable enough" for most purposes for small N.

Then p(x) = 1 / sqrt(2*pi*s0^2) * exp(- (x-m)^2 / (2*s0^2)), and P(X<0) = int from -inf to 0 p(x) dx

The colums are likely "Bets Placed" = N, "Theo. Win" = N*m0, and "Std. Deviation" = sqrt(N)*s0. From this you can retrieve m and s0, and calculate P(X<0) by integration of p(x).

Thank you so much!
Almost clear...
I have a question now how to get X ...."where X is the result after N bets" Is this an average result within +3s - -3s? Or this is OUTCOME X:

So for 500 hands 100000$ each X is = -500*100000*0.95= -47500000, Then (x-m)= -47500000-528954= -48028954

How to get exp from such huge number then?


...Or this is X outcome per 1 unit which is -0.95?

In this case I've got P(x) = 0.735596424

Should I integrate the whole equation in order to get P(X<0)?

Could someone just calculate the 1st row of the table to get this 0.399300...If I'm doing wrong...

Does the data in the rows stay the same if we change the headings

FROM Bets Placed ............ Prob. of Casino Loss ............Theoretical Win ........Standard Deviations.

TO..Bets Placed............... Prob. of Player Loss ............ Theoretical Loss ........Standard Deviations.

Quote: Winter

Thank you so much!
Almost clear...
I have a question now how to get X ...."where X is the result after N bets" Is this an average result within +3s - -3s? Or this is OUTCOME X:

So for 500 hands 100000$ each X is = -500*100000*0.95= -47500000, Then (x-m)= -47500000-528954= -48028954

How to get exp from such huge number then?


...Or this is X outcome per 1 unit which is -0.95?

In this case I've got P(x) = 0.735596424

Should I integrate the whole equation in order to get P(X<0)?

Could someone just calculate the 1st row of the table to get this 0.399300...If I'm doing wrong...

Back up. I'm not sure exactly what you're doing. The mean and the standard deviation up there are both in dollars; you don't have to multiply by 0.95, that's been done for you. The number that's important is the number of standard deviations the mean is from 0, and then you integrate from that number to infinity (or -∞ to the negative of that number) to get the chance of the casino losing.

So here's how you work out the first row of the table:

528954/2073667 ≈ 0.25508145714. For the casino to lose, the result has to be 0.25508145714 deviations below the mean.

Now you just have to consider the integral of the standard normal, which is equal to the infinite sum of (((-0.25508145714)^(2n+1))((-1)^n))/((2^n)*sqrt(2*pi)*n!*(2n+1)) from n = 0 to infinity if you want to do it by hand (you can establish an upper bound on a partial by the ratio test), but that's irrelevant because it's best to just plug it into a calculator, and mine is giving me their answer, now that I've found one that takes longer inputs.

EDIT: Oh, right, I see. You mean how to get those numbers in the first place. Okay, to get the mean and standard deviation from the number of bets, here's what you do: for the mean, multiply the number of bets by the bet size by 0.010579, which is .95*P(Banker win) - P(Player win). For the standard deviation, multiply the number of bets by the square of the bet size and 0.864679 (this is (.95+0.010579)^2*P(Banker win) + (1 - 0.010579)^2*P(Player win)), then take the square root. You can work out varying sizes by adding them all up, but don't forget in the latter case to square before adding them up then take the square root when you're done.

Thanks a million!..
All I need now is to find a proper online calculator :)

Hm I've got the first row result −.3989481634448608 with calculator... But later I've calculated this for 1500 hand and 35000...and result remained the same = −.3989481634448608...

For example:

As you said:

Lets calculate Casino Loss Probability for row with 20000 bets:

so 21,158,140/13,115,021= 1.613275342830179 (STDEV below the mean for casino to lose)


Series of (((-1.613275342830179)^(2*n+1))*((-1)^n))/((2^n)*sqrt(2*3.1415)*n!*(2*n+1)) by n on the interval from 0 to inf:
-.3989481634448608*'sum((-1)^n*1.613275342830179^(2*n+1)/((2*n+1)*2^n*n!),n,0,inf)
∑∞n=0.3989481634448608(−1)n(−1.613275342830179)2n+1(2n+1)2nn!=−.3989481634448608 ∑∞n=0(−1)n1.6132753428301792n+1(2n+1)2nn!

I'm using this calculator:

So, what I'm doing wrong again :D

That calculator's a joke. It's not giving you the answer at all, but a coefficient (namely, 1/sqrt(2*pi)); it's trying to find a closed form expression for the sum, and none exists. You probably want to leave the sum aside, unless you feel like doing it by hand, and go instead for calculator that does numeric integrations. You want the one from x to infinity, or -infinity to -x.

(Also note that you would want to start from 0, not 1.)

24Bingo, thanks for support and help. I highly appreciate it! Can you advice some good calculator to manage it via PM?