EV and SD for flat-betting 17,000 hands

matt21

Well-Known Member
#1
hi everyone, following on from the probability modelling work i did last week i wanted to do a quick analysis of how my actual results to date compared to a theoretical individual, let's call him Jack, that was flat-betting and playing perfect basic strategy for 17,000 hands.
For my game the house advantage is 0.55%

I have worked out the following -
p(win) for 1 hand = 0.49725 - P
p(loss) for 1 hand = 0.50275 - Q
# of hands played - 17,000 - N

Pushes are ignored since they are incorporated in the house advantage.

P(loss) - p(win) = 0.50275 - 0.49725 = 0.0055 = house advantage

Therefore, if flat betting and playing perfect BS (without index plays) Jack should expect to win 0.49725 x 17,000 = 8,453.25 hands

The standard deviation is calculated as follows (assuming results are normally distributed):

SD = (N x P x Q) ^ 0.5, thus
= (17000 x 0.49725 x 0.50275) ^ 0.5 = 65.1 hands

Thus confidence intervals for the number of winning hands would be as follows:
68% (8,388 - 8,518)
95% (8,322 - 8,583)
99% (8,258 - 8,649)

Interpreting the above, does this mean if Jack played numerous rounds of 17,000 hands, that he would be winning more than losing in approx 16% of the rounds (the 68 confidence interval is just outside the break-even point of 7,500 hands)?
Does it also mean that it would be highly unlikely (i.e. 0.5% probability) for Jack to be ahead by more than 149 units (8,649-8,500)

I would be keen for someone to comment on whether the above approach is correct or not.
 
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Sonny

Well-Known Member
#2
Your approach is essentially correct but the numbers are slightly different. Excluding ties, the probability of a win is about 46.7% and the probability of a loss is around 53.3%. The house edge is less than 46.7-53.3=6.52% because the payouts on splits, doubles and BJs are often more than one unit.

The variance of a single hand of BJ played with proper BS is 1.33 units. That gives us a SD of 1.15 per hand, so if we bet $1 on every hand the results will look something like this:

68% (-$243.86 to +$56.87)
95% 9-$394.23 to +$207.23)
99% (-$544.60 to +$357.60)

That gives you about a 26.7% chance of being ahead after 17,000 hands. There is a good article about this on Snyder's website:

http://www.blackjackforumonline.com/content/Blackjack_Basic_Strategy_Betting_And_Risk.htm

-Sonny-
 

matt21

Well-Known Member
#3
Hi Sonny, thanks for your reply here - appreciated! I think I understand some of the things that you are saying & also checked out the link to the article.

I understand how you get from the variance/hand of 1.33 to the SD of 1.15/hand. Could you please clarify the step in which you calculate the confidence interval for the 17,000 hands using the 1.15/hand SD rate?

What does surprise me is that there is still a 1/4 chance of Jack (the theoretical player) of coming out ahead after 17,000 hands - my gut feeling was that Jack's confidence would have shifted completely into the red.

P.S. Nice run thus far this week. Ahead 33 units in 12 hours of play, even after dropping a whopping 52 units in one session! 6 winning and 2 losing sessions. :joker:
 

Sonny

Well-Known Member
#4
matt21 said:
Could you please clarify the step in which you calculate the confidence interval for the 17,000 hands using the 1.15/hand SD rate?
Sure. The EV is 17,000*-0.0055 = -93.50. Your SD is sqrt(17,000)*1.15 ~ 150. Your range of results for one SD is roughly:

-93.5-150 = -243.5
-93.5+150 = 56.5

You can expand that by multiplying the SD by 2,3,4 etc.

matt21 said:
What does surprise me is that there is still a 1/4 chance of Jack (the theoretical player) of coming out ahead after 17,000 hands - my gut feeling was that Jack's confidence would have shifted completely into the red.
17,000 hands is nothing. That's only about 170 hours of play. It would take about 58,500 hands (585 hours) to overcome just one SD and about 234,000 hands (2,340 hours!) to overcome 2 SDs. Since the house advantage is so tiny it takes a loooong time to approach the long run.

-Sonny-
 

Kasi

Well-Known Member
#5
matt21 said:
What does surprise me is that there is still a 1/4 chance of Jack (the theoretical player) of coming out ahead after 17,000 hands - my gut feeling was that Jack's confidence would have shifted completely into the red.
That's what everyone thinks, I think lol.

It probably assumes he has enough roll to not go broke before 17000 hands.

I don't know how it differs from Sonny's approach, if at all, but I use this formula from the wizardofodds website

=normsdist((r+t*h+0.5)/(t^0.5*1.16)).

Where, r = the number of units won or lost, t = the number of hands, and h = the house edge.

So it'll tell you how many SD it is.
 

matt21

Well-Known Member
#6
Sonny said:
Sure. The EV is 17,000*-0.0055 = -93.50. Your SD is sqrt(17,000)*1.15 ~ 150. Your range of results for one SD is roughly:
-93.5-150 = -243.5
-93.5+150 = 56.5
-Sonny-
all right, got you. that makes sense now. thanks for the explanation!


Sonny said:
17,000 hands is nothing. That's only about 170 hours of play. It would take about 58,500 hands (585 hours) to overcome just one SD and about 234,000 hands (2,340 hours!) to overcome 2 SDs. Since the house advantage is so tiny it takes a loooong time to approach the long run.
-Sonny-
It always depresses me a little when i hear statement like yours saying "17000 hands is nothing". You are of course dead right - it's just that it has taken me 7-8 months to log 250 hours of play! In contrast, the casino plays for 24 hours per day every day so they obviously reach the long run a lot quicker!
However I am planning to log up my hours significantly faster in coming months so let's see whether i can maybe hit that 1 SD threshhold by the end of the year.

Thanks again for your comments.
 

matt21

Well-Known Member
#7
Kasi said:
I don't know how it differs from Sonny's approach, if at all, but I use this formula from the wizardofodds website
=normsdist((r+t*h+0.5)/(t^0.5*1.16)).

Where, r = the number of units won or lost, t = the number of hands, and h = the house edge.
So it'll tell you how many SD it is.
So using this formula we are able to pin-point exactly how many standard deviations we are from EV.

If we use my figures - of course we are now no longer dealing with flat-betting since i am using a 1-12 and sometimes 1-20 spread across the various true counts as well as 34 index plays.

r=+288
t=17,115 (244.5 hours assuming 70 hands/hour)
h=0.0055

My answer came out as 2.46 i.e. my actual result is 2.46 standard deviations from the EV. 2 SD equates to the 95% confidence interval and 3 SD equates to the 99% confidence interval, so 2.46 falls somewhere outside the 95% CI but within the 99% CI. Again, this is of course just putting my non-flat betting results into Jack's theoretical analysis but i think, this does to indicate that i am doing all right.

Thanks for pointing out this formula Kasi!
 

Kasi

Well-Known Member
#8
matt21 said:
So using this formula we are able to pin-point exactly how many standard deviations we are from EV.

If we use my figures - of course we are now no longer dealing with flat-betting since i am using a 1-12 and sometimes 1-20 spread across the various true counts as well as 34 index plays.
Again, this is of course just putting my non-flat betting results into Jack's theoretical analysis but i think, this does to indicate that i am doing all right.
Well, I may have mis-led you a little bit lol. The answer you get after putting the units in would maybe better be stated as the chances of losing that much or more (I think) - not exactly that amount.

And it is only for flat-betting - no way should you compare a CC's results to it, if that's what you were doing. For one the HA would change, for another so would the stand dev. That's when it's maybe time for a sim to tell you your avg bet units, adv %, and stan dev in units.
 

Kasi

Well-Known Member
#9
matt21 said:
It always depresses me a little when i hear statement like yours saying "17000 hands is nothing". You are of course dead right - it's just that it has taken me 7-8 months to log 250 hours of play!However I am planning to log up my hours significantly faster in coming months so let's see whether i can maybe hit that 1 SD threshhold by the end of the year.
I'm not sure I understand what you are doing - are you counting and spreading, whether for real or practice?

If you are, you should/could know your "long-run" even before you play.
It's possible 17000 hands could be the long run for a card-counter. If the card-counter is dead-even after 17000 hands it's a neg SD result. If a flat-betting BS player is, it's a +SD result.

Can you measure your results (I guess these 200+ units) now for what you've been doing for these 250 hours? How much are they above or below expectation?
 

matt21

Well-Known Member
#10
Kasi said:
I'm not sure I understand what you are doing - are you counting and spreading, whether for real or practice?

If you are, you should/could know your "long-run" even before you play.
It's possible 17000 hands could be the long run for a card-counter. If the card-counter is dead-even after 17000 hands it's a neg SD result. If a flat-betting BS player is, it's a +SD result.

Can you measure your results (I guess these 200+ units) now for what you've been doing for these 250 hours? How much are they above or below expectation?
Hi Kasi. These are real results for real casino playing. I am counting and spreading :grin: but wanted to try to put my results in the light of a theoretical player that was simply flat betting, to see how my actual results compare against that.

I have not (yet) got into detailed analysis for determing my long-run threshhold (that's called 'N' or something like that?).

I think it would be difficult for me to calculate a EV for the actual time played because i have played a mix of 4D and 6D games, have encountered a range of 70-90% pen (but the majority in the 75-83 range) and have also changed my bet variation over the time that I ahve played.

However now that i think i understand the EV and SD calculations better, i should be in a position to compare my future play against EV and SD.

Very nice day today by the way. Scoped out two new playing venues and made 50 units in three hours without TC ever going above +2 but with nearly all the doubles, splits and black-jacks just falling my way :laugh: - such days are much nicer than the two roller-coaster ride days i had earlier in the week!!
 

Kasi

Well-Known Member
#11
matt21 said:
I think it would be difficult for me to calculate a EV for the actual time played because i have played a mix of 4D and 6D games, have encountered a range of 70-90% pen (but the majority in the 75-83 range) and have also changed my bet variation over the time that I ahve played.
Exactly - you got it! :) Get's a little difficult a few hundred hours later lol.


matt21 said:
However now that i think i understand the EV and SD calculations better, i should be in a position to compare my future play against EV and SD.
Very nice day today by the way....
So, then, why not begin with analyzing today's play then while it's still maybe just the same game played the same way and you can estimate hands played to some degree? How lucky were you today?

The future is now lol.

I'm a tough guy lmao - don't give me this "think" and "should" stuff. Give me "do" and "am" :)
 

matt21

Well-Known Member
#12
Kasi said:
So, then, why not begin with analyzing today's play then while it's still maybe just the same game played the same way and you can estimate hands played to some degree? How lucky were you today?
I get you. So I was just about to tell you exactly how lucky I was, by subtracting my actual result from the EV - bearing in mind that my EV for the first two hours was lower as I was playing with 2-4 others at a $2 table - before playing a $5 table on my own for the last hour. The EV changes immediately because of the number of hands that I would be playing. However I just realised that I still need to fine-tune my probability model for the fact of me playing 2 hands at all times.

In any case my overall 3-hr EV would be likely be between 3 and 4 units - thus I was lucky to the tune of 49-50 units.

That is one of the troubles with my model. My EV changes constantly depending on the number of players at the table AND the varying deck penetration.

All in all, last week was a great one - up $990 in 21.5 hours.


Kasi said:
I'm a tough guy lmao - don't give me this "think" and "should" stuff. Give me "do" and "am" :)
I fully agree with you on this. I will try to give you as much "do" and "am" as possible.
 
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Canceler

Well-Known Member
#13
Quick, before he sees this!

matt21 said:
In any case my overall 3-hr EV would be likely be between 3 and 4 units - thus I was lucky to the tune of 49-50 units.
Preview of coming attractions:

Kasi's probably gonna :whip: you for not expressing this in standard deviations!
 

matt21

Well-Known Member
#14
Canceler said:
Preview of coming attractions:

Kasi's probably gonna :whip: you for not expressing this in standard deviations!
Oh oh!
Okay SD for 3 hours of play was approx 68 units. So the result of +52 fell within one standard deviation.
 
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Kasi

Well-Known Member
#15
matt21 said:
I was just about to tell you exactly how lucky I was, by subtracting my actual result from the EV - bearing in mind that my EV for the first two hours was lower as I was playing with 2-4 others at a $2 table - before playing a $5 table on my own for the last hour. The EV changes immediately because of the number of hands that I would be playing. However I just realised that I still need to fine-tune my probability model for the fact of me playing 2 hands at all times.
Thanks for replying. Seriously. I was afraid I may have been too blunt, as I often am, so, don't ever take it personally :)

Here's my thoughts - if the same game, spreading to the same unit spread at the same counts, whether they be $2 units or $5 units, whether there be 2 or 7 players, you only make so many units a hand and have the same SD per hand in units. Naturally dollar EV and dollar SD would change. Obviously so would ROR at higher dollar units with the same roll. But units per hand would not change.

So, segregate results accordingly.

I hope your "probability model" is based on a sim? lol.

You seriously always playing 2 hands, even in neg counts? Are you wonging-out maybe?

What is this probability model? - I hope you are at least using your avg bet units in it and not your min unit especially if you are using that 1.15 SD.

We'll pick just one game to start, pretend you will play only that game that way tomorrow, to keep it simple - say $5 unit, pick a spread and game and pen, playing one hand heads-up. What's your probability model say then?

Feel free to pursue to this publicly or privately if you want. No big deal.
 

matt21

Well-Known Member
#16
Kasi said:
Here's my thoughts - if the same game, spreading to the same unit spread at the same counts, whether they be $2 units or $5 units, whether there be 2 or 7 players, you only make so many units a hand and have the same SD per hand in units. Naturally dollar EV and dollar SD would change. Obviously so would ROR at higher dollar units with the same roll. But units per hand would not change.
...
Feel free to pursue to this publicly or privately if you want. No big deal.
Hi Kasi, of course I would like to pursue this discussion further with you. In fact i am finding this very interesting (and of course valuable and helpful!).

Re EV - i always talk of EV in terms of EV per hour rather than per hand, and that is of course impacted by the number of hands played per hour, which is in turn impacted by how many players are sitting at the table.

Ok so I will have different versions of the model, eg 4D games, 6D games, bet variations (e.g. 1x$2, 2x$2, 1x$5 etc etc) - note that at the $2 table i was still playing with $5 min bet.

Yes the model is based on a sim (from CVD) - i entered the game variables into the CVD and then identified the distribution of true counts across the true count groups i.e. the % of time that each true count actually occurred during the sim. I then fed these %'s into my model ranging from +8 to -8.

Yes, sometimes i play 2 hands even during negative counts. I have provided for this impact in my model. I do wong out if count gets very bad.

I guess, I will need to forward you the spreadsheet so that you can have a look at it and see what you think, and then maybe we can chat online together.
 

sagefr0g

Well-Known Member
#17
fact checker on ilse 5

Quote:
Originally Posted by Kasi
Here's my thoughts - if the same game, spreading to the same unit spread at the same counts, whether they be $2 units or $5 units, whether there be 2 or 7 players, you only make so many units a hand and have the same SD per hand in units. Naturally dollar EV and dollar SD would change. Obviously so would ROR at higher dollar units with the same roll. But units per hand would not change.
matt21 said:
Hi Kasi, of course I would like to pursue this discussion further with you. In fact i am finding this very interesting (and of course valuable and helpful!).

Re EV - i always talk of EV in terms of EV per hour rather than per hand, and that is of course impacted by the number of hands played per hour, which is in turn impacted by how many players are sitting at the table.
just to help out a little maybe or maybe confuse the issue as the case may be as far as how many players at the table in one instance and how many players for another instance. here's two canned sims same game. note the custom bets in the 7player sim were forced to match the spread and ramp of the spread and ramp of the heads up player sim.
so for the two instance of heads up and 7player if the spread and ramp stay the same the the unit winrate and std dev per hand is pretty darn close not exact but close. but i guess then there is the question of how these two canned sims were produced as far as figuring tc's and all.
 

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Kasi

Well-Known Member
#18
sagefr0g said:
just to help out a little maybe or maybe confuse the issue as the case may be as far as how many players at the table in one instance and how many players for another instance. here's two canned sims same game. note the custom bets in the 7player sim were forced to match the spread and ramp of the spread and ramp of the heads up player sim.
so for the two instance of heads up and 7player if the spread and ramp stay the same the the unit winrate and std dev per hand is pretty darn close not exact but close. but i guess then there is the question of how these two canned sims were produced as far as figuring tc's and all.
Yeah, you're right - I shouldn't have included a different number of players. That makes it a (slightly) different game. I think maybe lol.

Of course you will always get slightly different results running the same sim twice, I think. After all, every 400,000,000 hands is likely a little different than the next 400,000,000 hands lol.

But maybe it could also be stuff like playing heads-up, when you bust the dealer takes no more cards. With 7 people he almost always will unless all 7 people bust.

It's not like one can run too many sims for whatever one may expect to run into.

I just meant, once you have a sim, it shouldn't make any difference - like the WW game, change the roll to $10K or $1000K, and, or the dollar unit to $1 or $1000, the units per hand etc won't change. How could they when you are betting the same unit spread at the same time in the same game?
 

sagefr0g

Well-Known Member
#19
Kasi said:
Yeah, you're right - I shouldn't have included a different number of players. That makes it a (slightly) different game. I think maybe lol.

Of course you will always get slightly different results running the same sim twice, I think. After all, every 400,000,000 hands is likely a little different than the next 400,000,000 hands lol.

But maybe it could also be stuff like playing heads-up, when you bust the dealer takes no more cards. With 7 people he almost always will unless all 7 people bust.

It's not like one can run too many sims for whatever one may expect to run into.

I just meant, once you have a sim, it shouldn't make any difference - like the WW game, change the roll to $10K or $1000K, and, or the dollar unit to $1 or $1000, the units per hand etc won't change. How could they when you are betting the same unit spread at the same time in the same game?
yeah on the changing up your units thing and probably a not really significant differance as to the number of players sorta thing. i'm gonna try and do a couple of sims on the number of players thing that maybe shouldn't have to much variation i guess.
but yeah i guess it could happen you run a sim, maybe run it again and not everything is the same. hopefully close but no cigar lol. maybe in part for the reasons in these posts:
http://www.blackjackinfo.com/bb/showpost.php?p=94782&postcount=37
http://www.blackjackinfo.com/bb/showpost.php?p=94931&postcount=51
http://www.blackjackinfo.com/bb/showpost.php?p=95139&postcount=506
weird stuff. and i don't even begin to understand how ever much of it.
like when i was in chemistry class and the professor was kind enough to explain to us the schrodinger equation and have us working solutions for the hydrogen atom with it even though half of us had never had even statistics or probability 101.
 

matt21

Well-Known Member
#20
bump...
rather than start a new thread i thought i ask the question since it relates to the whole topic of SD etc.

calculating standard deviation (SD):

Assume that SD for one shoe is 46 units

Then for 2 shoes, the SD will be:
SQRT(2) * 46
=65

For 4 shoes, the SD would be:
SQRT(4) * 46
=92

But now assume that we know the SD for 2 shoes and wanted to know the SD for 1 shoe - do we do it this way?
SD (2 shoes) = 65

65 / (SQRT(2)
= 46

is that the correct approach?

Thanks in advance,
Matt
 
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