Std Deviation

[Z]

What is the applicable std deviation calculation for bj?

is it sqrt(p*q/n)?

where
p=50 - ("est house edge"/2)
q=1 - p
n=number of hands

So for house est edge .16% for 500 hands, it would be
sqrt(.5008*.4992/500)=2.24%

Is that right?

For unit bets +/- 3 std devs from mean (-.8 unit) about +/- 35 units for 99% confidence?

This all seems reasonable, is it?

Thanks,
Z

 

[The Mayor]

>What is the applicable std deviation calculation for bj?

>is it sqrt(p*q/n)?

NO

>where
>p=50 - ("est house edge"/2)
>q=1 - p
>n=number of hands

No, this is for a binomial experiment. Because of pushes, splits, and double downs, this does not apply to blackjack.

>So for house est edge .16% for 500 hands, it would be
>sqrt(.5008*.4992/500)=2.24%

>Is that right?

It is a correct computation, just not applicable to blackjack.

>For unit bets +/- 3 std devs from mean (-.8 unit) about +/- 35 units for 99% >confidence?

As far as I know, SD and EV are only known via simulations for BJ.

--Mayor

 

[zengrifter]

"As far as I know, SD and EV are only known via simulations for BJ."
--------------------

StDv and Ev can be calculated by hand PROVIDED that one use a 'frequency distribution' table, such as those provided in the ASnyder 'Beat The X Deck' series. zg

 

[Adam N. Subtractum]

"Root mean squared deviation of Blackjack"

I'm not sure what you're looking for here, but I'll try to help you out a with a few comments.

If all we need is the Standard Deviation per hand, we can get a good idea from the numbers in table 85 of Wong's PBJ, since the square root of Variance is the Standard Deviation. We see in table 85 the Variance of a single hand of Blackjack ranges from 1.20 to 1.32 (this can actually vary more with rarely found rules), so the Standard Deviation of a single hand of Blackjack will range from approximately 1.1 to 1.15.

Now to calculate the SD over a number of hands, we simply equate Variance * sqrt(n), where n is the number of hands played. So for 500 hands:

1.1 * 22.36 = 24.596

1.15 * 22.36 = 25.714

We see we can expect a swing of up to 24.6 to 25.7 units in one Standard Deviation, which will occur 68.3% of the time. We will see swings of up to double that, an additional 26.7% of the time, and see swings of up to 3 SD's an additional 4.7% of the time on top of that. This tells us that, for our example, we will lose more than 73.8 to 77.1 units, less than .3% of the time.

It's important to note, that like the even money Kelly equation (ev*BR) used by some, this method maybe sufficient for Wong in/Wong out players, but will undoubtedly underestimate the Risk of the play-all Bankroll.

As the Mayor stated a sim really is necessary to get an accurate figure, but I'm sure we can at least get in the ballpark.

Another method is to DIVIDE the SD by the square root of hands played, to get SD in a percentage. For our previous example:

1.1 / 22.36 = 4.919%

1.15 / 22.36 = 5.143%

We could then multiply this with our total action to equal the amount of $$$ in one SD swing. As zg suggested we would need frequency of advantage (or more ideally, frequency of True Count) distributions for the game in question, so we could calculate the frequencies of each of our bet sizes in order to sum to our total action per the 500 hands. This figure would be our $SD$/500 hands (Standard Deviation in cash, per 500 hands), and we could multiply this number by the square root of the number of 500 hand sessions played to compute the SD for any number of sessions played.

Forgive me if I have made any stupid mistakes...it is wayyy past my bedtime |-)

ANS

 

[Rob McGarvey]

Very Clearly Explained

Anyone can undeerstand SD the way you have presented it. Now you get to bed! I'm going to have my morning Java and trounce a few online bj games before the lil' guys get up.

 

[zengrifter]

I'm still wrestling with it (not my area)

 

[Z]

Thank You and a little more...

Thanks guys. Amazon just delivered Pro BJ by Wong and Griffin's Theory of BJ so I am better equipped for these questions now.

First, I made a big error with applying binomial probability to BJ as follows: In BJ edge is stated as a percentage of total money wagered. This is why I divided the edge by 2 before applying the std dev calculation. But I neglected to multiply my end result by 2 to bring it back to BJ terms. So my result of around 2.25 should have been multiplied by 2 to 4.5% as applicable to BJ. I know that this is not exact but it seems to land in the same ballpark, which is close enough for me. I'll take the larger value, call it a nice even 5% for BS play and I am happy with that.

In any event, it is highly unlikely (less than .5%) to reach +/- 75 units in 500 hands.

Thanks again ANS and Mayor for your valuable inputs, and I hope that this 5% ballpark for std dev may be helpful to others as a gauge for possible outcomes with relatively small number of hands.

Z

 

[Rob McGarvey]

Re: I'm still wrestling with it (not my area)

Basically all of the background math behind a count system and bet sizing. Most counters don't have to understand it, but it is reasuring to know there is a solid foundation under our feet mathematically speaking. All interesting stuff, but can end up being more of a distraction than anything else. I'm more interested and focused on the billions of dollars being played online right now. So it is quickly becoming "not my area" too.

 

[zengrifter]

Gee, now I get it!