Per-Hand Expectation and Risk of Ruin

[nismo]

I'm reading Blackjack Attack, and am trying to do some calculations of my particular betting style, but am stuck on my "per-hand expectation." For those of you that own the book, I'm referring to the charts on pg 116 - 121. How do I calculate this "expectation," or am I used to it being called something else? Am I just having a brain lapse here? :whip:

Anyway, for those of you that don't have the book, it's really nice. Schlesinger gives several pages of charts that list the RoR for many of the SD & "expectation" combinations of 200, 300, 400, well you get the picture, up to 1000 unit bankrolls!

 

[callipygian]

I'm reading Blackjack Attack, and am trying to do some calculations of my particular betting style, but am stuck on my "per-hand expectation."

I have never read the book, but I think I can help.

You can calculate your per-hand expectation by multiplying your bet spread with the advantage at a count and the probability that count will come up.

Mathematically,

expectation/hand = Sum(i = -inf to +inf) {bet(i)*EV(i)*percent(i)}

So, let's say you Wong out at Hi-Lo TC -2 on an 8-deck shoe and a baseline house edge of 0.5% (to make the math simple), and spread 1-5 evenly from TC +1 to TC +5. We'll assume advantage is linear with count (it's close enough) and that TC's of <-5 and >+5 don't show up.

Your per-hand expectation should be:

expectation/hand = Sum(i = -5 to +5) {bet(i)*EV(i)*percent(i)}

expectation/hand = 0*(-0.030)*(0.01) + 0*(-0.025)*(0.02) + 0*(-0.020)*(0.04) + 0*(-0.015)*(0.09) + 1*(-0.010)*(0.19) + 1*(-0.005)*(0.30) + 1*(0)*(0.19) + 2*(+0.005)*(0.09) + 3*(+0.010)*(0.04) + 4*(+0.015)*(0.02) + 5*(+0.020)*(0.01) = +0.0009, or 0.09%.

(this is very easy to calculate in an Excel spreadsheet, don't be scared by the length of that equation)

Note that this is different from EV/hand, which is expectation/hand divided by your average bet. You can calculate your average bet in exactly the same manner: your bet at a count times the probability that count comes up.

average bet = Sum(i = -inf to +inf) {bet(i)*percent(i)}

average bet/hand = 0*(0.01) + 0*(0.02) + 0*(0.04) + 0*0.09) + 1*(0.19) + 1*(0.30) + 1*(0.19) + 2*(0.09) + 3*(0.04) + 4*(+0.02) + 5*(+0.01) = 1.11 units.

So your EV/hand on this play is +0.0009/1.11 = +0.0008, or +0.08%.

 

[nismo]

Wow, thanks for that great response! :toast:

One question though, for the expectation/hand formula example you provide:

expectation/hand = 0*(-0.030)*(0.01) + 0*(-0.025)*(0.02) + 0*(-0.020)*(0.04) + 0*(-0.015)*(0.09) + 1*(-0.010)*(0.19) + 1*(-0.005)*(0.30) + 1*(0)*(0.19) + 2*(+0.005)*(0.09) + 3*(+0.010)*(0.04) + 4*(+0.015)*(0.02) + 5*(+0.020)*(0.01) = +0.0009, or 0.09%

What does the third number of each combination represent (ie, 0*(-0.030)*(0.01))? It is percent(i), but what is this exactly? Is this the frequency of each hand? Thanks for your patience, I'm relatively new to counting as I've only been at this for about a year now. I feel like I have 50 years worth of blackjack development to catch up on and sometimes the mathematical principles behind AP still :bomb: me!

 

[callipygian]

Is this the frequency of each hand?

Yes; it will vary by the number of decks used and the penetration. I gave a generic figure which roughly corresponds to 75% penetration on an 8D game. I only have numbers of 4D, 6D, and 8D games because that's what I usually play - I'm sure someone has a count distribution for 2D and 1D games.

You can calculate it yourself pretty easily with an Excel spreadsheet, but it's time-consuming to generate a good sample size. It took me ~90 min of refreshing and cutting and pasting to get 100,000 shoes.

 

[nismo]

I'm actually interested in the frequency numbers for a 6D game (although I am curious what effects on RoR switching to a 4D game would have, one would think RoR would go down but by how much?). So... what, to flog a horse, that if not dead is at this point in mortal danger of expiring, would be the 6D numbers (assuming 75% pen)?

Or even better, could you teach a man to fish? How can I calculate them myself? Is there public access to Excel files already formatted for such calculations, so that one can simple enter their data or is this something that you've made yourself?

In the spirit of mutually beneficial exchanges :grin:, I've found the frequency numbers for a 1D, 75% pen game, if you're curious. These are provided by Steve Jacobs in his paper, "Optimal Betting," which is quite fascinating (this could all be old news to you). In it, he discusses expected utility, where one betting scheme can be rated against another:

http://www.bjmath.com/bjmath/Betsize/sjopt.htm (Archive copy)

-5 .065
-4 .030
-3 .055
-2 .070
-1 .100
0 .200
+1 .095
+2 .075
+3 .050
+4 .045
+5 .040

 

[rukus]

Or even better, could you teach a man to fish? How can I calculate them myself? Is there public access to Excel files already formatted for such calculations, so that one can simple enter their data or is this something that you've made yourself?

if you are in this for the long haul, QFIT's CVCX or CVData cost some coin but are the standard and well worth the money, and more importantly can provide the TC frequencies you desire. if this is just a one-off thing, take a look at powersim which is a free simulator and can do the same thing with a simple manipulation of the output.

 

[Sonny]

Is there public access to Excel files already formatted for such calculations, so that one can simple enter their data or is this something that you've made yourself?

Ask and ye shall receive:

http://www.blackjackinfo.com/bb/showthread.php?p=15578

-Sonny-

 

[nismo]

if you are in this for the long haul, QFIT's CVCX or CVData cost some coin but are the standard and well worth the money

I'd very much like to get CVCX along with Casino Verite... but I'm a mac user, and from what I understand, there are no good blackjack programs for mac right now. This leaves me to old fashioned pen and paper for the time being..

http://www.blackjackinfo.com/bb/showthread.php?p=15578

-Sonny-

I could kiss you! :grin:

 

[rukus]

I'd very much like to get CVCX along with Casino Verite... but I'm a mac user, and from what I understand, there are no good blackjack programs for mac right now. This leaves me to old fashioned pen and paper for the time being.

use apple's Boot Camp. i too use a mac sometimes and i just use Boot camp to boot into windows and run QFIT's products (in addition to excel!). google apple boot camp.

there are programs that allow you to run windows software without booting into a copy of windows but 1. i do not know much about them and 2. i hear they are buggy and sometimes crash even the mac portion of your computer.

 

[Kasi]

How do I calculate this "expectation,"...

Not only that EV but also, like you say, the SD too lol. As you suggest, that's the key - can't use the tables without them lol.

About the only way I know of to generate the frequencies and advantages and SD by count is to run a sim.

They will change not only with decks and pen and rules but also with counting system, use of indexes, whether TC's maybe are rounded or floored etc.

You mention 4D maybe lowering ROR vs 6D but keep in mind you can more or less bet any game with any ROR you want. Unit sizes may change, or spreads or playing-all vs back-counting or wonging out somewhere may change etc but ROR can stay about the same. Why not play every game with a ROR you are comfortable with and let your EV change? Or, if you want, keep EV the same but then ROR will change, assuming same roll lol.

You can get an idea of frequencies and advantages (and SD by count) from BJAIII CH 10 tables, given the assumptions he used, (like Hi-Lo, flooring, I18 indexes I think) but they lump all counts below 0 together. If you can reproduce with a spreadsheet the right side of his tables given the first 3 columns on the left-side and using the spreads on the left side, you've learned how to fish

Since you liked those tables so much, here's a sheet of them expanded to 5SD on the left and up to 2000 units with the underlying formula used to generate them. But with the formula, you won't need the tables when EV is in a gap or SD exceeds the range lol.

 

[Kasi]

Is there public access to Excel files already formatted for such calculations, so that one can simple enter their data or is this something that you've made yourself?

Fwiw, here's a sheet for Don's Table 10.4 on page 216 for a play-all practical1-16 spread (line 6) with $10K roll and $5 unit like he assumes.

Just to give you an idea of the calculations once you have the the freq, adv and SD info.

Word of warning - this version will not work perfectly for back-counting, wong-out scenarios. Should be fine for play-all scenarios.

Just intended to give you an idea - along the same lines Callipygian suggested.

 

[nismo]

Not only that EV but also, like you say, the SD too lol. As you suggest, that's the key - can't use the tables without them lol.

About the only way I know of to generate the frequencies and advantages and SD by count is to run a sim.

Unless I'm mistaken,

SD = 1.1√number of hands played
EV =sum of [(probability of win x amount) - (probability of loss x amount)] for each count played

Now advantage and, obviously, frequencies I do not know how to calculate. I am forced to used other peoples' numbers to plug into probability of win/ probability of loss for each count when finding EV. Probability of a win is just = .5 + advantage.

You mention 4D maybe lowering ROR vs 6D but keep in mind you can more or less bet any game with any ROR you want.

I was assuming keeping all other variables constant, ie just switching from 6D to 4D. But you're right, with a more complete understanding of the odds, one could taylor specific play for each game!

Since you liked those tables so much, here's a sheet of them expanded to 5SD on the left and up to 2000 units with the underlying formula used to generate them. But with the formula, you won't need the tables when EV is in a gap or SD exceeds the range lol.

You're so good to me!

 

[Kasi]

Unless I'm mistaken,

SD = 1.1√number of hands played

That 1.1 (units per dealer upcard) assumes flat-betting. Simplified.

If you're betting different units at different TC's, like CC's do, even though that 1.1 is more or less constant at each TC if you had flat-bet each TC, it can change, on average, wildly with spreads.

Check out why in Don's Tble 10.4 sheet why SD per round is 4 units, $20 in dollars when min unit is $5, and avg bet is $10 or 2 units.

Yet the SD at each TC only varies from 1.13 to 1.15 (your 1.1) across all TC's.