Standard deviation and expect value S&L

[Cardcounter]

How standard devaition works for the short and long term!
Lets look at standard deviation and its effect on short term results

Lets say you are cardcounter with a 1% average advantage and you play a 100 hands at an average of a $100 a hand. Standard deviation is equal to the square root of the number of hands played times a 110% of the amount bet. So one standard deviation can produce results bad enough to put you a loser in such a short period of time.


#of hands expected value of play is standard devaition is
100 $100 10*110=$1,100+or-
10,000 $10,000 100*110=$11,000
1,000,000 $1,000,000 1,000*$110=$110,000
100,000,000 $100,000,000 10,000*$110=$1,100,000
10,000,000,000 $10,000,000,000 100,000*$110=$11,000,000

The point is even though you could be a loser after 10,000 hands if you truely had an advantage after 1,000,000 hands you will never be a loser. At 10,000 hands the odds of losing a $1,000 and winning $21,000 are the same. At a 100 hands the odds of losing $1,000 and winning $1,200 are the same.
At a 1,000,000 hands of playing with a 1% advantage the odds of winning $890,000 to $1,110,000 are the same even if you had terrible variance and where 3 standard deviations lower than where you should be at a 1,000,000 hands you would still be $670,000 ahead which would be just as likely as being $1,330,000 ahead! Thats why short term flucktion might suck but long term it is golden.
The point is the larger the number of hands the smaller the relative standard deviation is. At a 100 hands the standard deviation was 1,100% of expected value. At a 100 million hands the standard deviation shrank to 1.1% of expected value.
If you play 10,000 hands or less you have a far shot at losing money even if you have a 1% disadvantage. You certanly have a really far shot at losing money playing only a 100 hands.

 

[eps6724]

THIS makes sense. Not being exactly 'strong' figuring this stuff out, but laid out like this I understand!

Thanks!
-EPS

 

[EyeHeartHalves]

good explanation of SD

I get my win rate/100, SD/100 and a whole lot of my other stats from Professional Blackjack Analyzer for Windows. Then I break out my old stats textbook when I need more complex questions answered such as, "How likely am I to make $1 or more on my next trip?"

(Btw: I call 100 hands a block. Sorry, but in Jerzee that's more like an hour and a half--not an hour.) My simulated win/block is $55, my simulated SD/block is $1,225, my simulated average initial bet (AIB or "u") is $57, and my simulated risk of ruin (RoR) is 1% for 1,000 AIBs.

Personnally, I don't believe SD pertains very much to the long-term. Nothing you said is wrong. It's just that what you're saying is that after a million or so hands (or a couple decades of playing in some cases), you'll be up! However, this assumes an infinite bankroll. I want my "lifetime bankroll" to have less than a 1% RoR. In my case that's at least $57,000 but I don't actually have that much. However, "I play to this much" (I'll get back to this concept.), pretending I have this much, and at this point in my carreer, I'm trying to save and am prepared to put up twice this much.

SD is very valuable to me. I use it for the short term measurement of risk. Let's say I plan to play 1000 hands in a weekend trip. That's 10 "blocks." My SD/block is 1225. However, my SD/trip is not 12,250 because that is not the nature of standard deviation. Like we established, you're up after a million hands because the size of the SD becomes exponentially smaller as the number of trials increases.

So here's how I figure my "Trip Bankroll." To get the SD/multiple trials (in this case it's 10), you take the square root of the "amount of trials less one". This is otherwise pronounced, "the square root of n minus one." In my case it's the SQROOT of ten minus one. (The answer is 3.) You then multiply this factor by your SD/trial ("block"). So, in my case it's 3 X 1225 or for the sake of conservatism and rounding--$3,700.

The problem with that is that it's just 1 SD/trip. I personnally like to have three times my trip SD per trip. One SD is very risky, two SD is a little risky and three SD is very low risk. (On the bright side, reality does seem to be better than simulation because I've never had to dip into even a second SD on any of my trips.)

Getting back to what I was saying about RoR. After I simulate my stats and get my lifetime bankroll, I double checked my bet spread with the "Kelly Criterion." Kelly betting simply means that you multiply your percent advantage by your bankroll (lifetime) to decide each and every bet you make. This procedure is both impractical and too risky for bj purposes.

First, Kelly says you'd bet zero with a 0% advantage and you'd bet a negative amount against a House Edge. Well, the latter is impossible and the former idea is difficult without team play. Thus, CC's chose to use what's called Half-Kelly where you'd figure your Kelly bet normally but than Halve every one of them when it's greater than your minimum bet. Unfortunately, this was discovered (by Arnold Snyder and others) to still be too risky.

This brings us to the other risk related problem. Your initial bet does not always equal your total bet for the hand. This is especially true at higher counts where you do actually have an edge. For example, (rounding down 2k) let's go back to my bankroll of $55,000 and the double checking of my bet spread. At a halves TC of +4, I have about a 2% edge over the house for said hand. According to Half-Kelly, I should be betting $550 (55k X 0.02 / 2). However, in a DA2-DAS game with a TC=+4, it's extremely likely that I'll have a DD or split or both which can quickly escalate to "Double Kelly" (or double trouble because you don't only give back your edge with double Kelly, you may also increase the House Edge!)

Thus, we arrive at Snyder's (and others) idea of "Approximate Quarter Kelly." (So, in the above example, I'd be betting $275. And since all my bet spreads with this hypothetical $57,000 were extremely close to Quarter Kelly, I passed the test and was ready to go from the computer simulation program to the dining room table.) I say approximate because of the "practical problem" I mentioned above with reguards to Kelly betting. You can't do it in any casino. You'd end up having to bet units that don't exist such as pennies and dimes and such. Furthermore, you can't be worried about all the things a CC has to worry about and then be adjusting your bet spreads by red chips every time you have a change in your bankroll.

I like the idea of reconsidering my bet spreads only after 20% shift of my bankroll. And as I showed in the above example, my trip BR happens to be about 20% of my lifetime BR so I'd never have to worry about changing my spreads mid-trip unless it's a long trip.

Well, sorry for such a long post. I was bored and I hope this may help some of the new CCs out there. Most importantly:

Remember that RISK OF RUIN and STANDARD DEVIATION are two completely seperate ways of helping you decide upon your bankroll. And the KELLY CRITERION can help you decider upon bet spreads.

[email protected].

 

[Kasi]

Well, sorry for such a long post./QUOTE]

Holy crap! :grin:

Something I think I actually understand and makes complete sense.

Absolutely no apologies necessary.

 

[Sonny]

I want my "lifetime bankroll" to have less than a 1% RoR. In my case that's at least $57,000 but I don't actually have that much. However, "I play to this much" (I'll get back to this concept.), pretending I have this much, and at this point in my carreer, I'm trying to save and am prepared to put up twice this much.

You can use the Renewable Bankroll ROR formula for that. It is a ROR formula that is tweaked to include the regular installments you are making to your bankroll:

B = B0 ( 1 + sqrt(1 + 4 I N0/(k B0)) ) / 2

B0 is your actual present bankroll,

I is your non-gambling income per round (or the non-gambling income per month divided by the number of rounds you play per month),

N0 is the doubling time in rounds,

k is your Kelly factor (use 0.7071 if you desire 5% RoR and no unit resizing),

B is your effective bankroll that you are allowed to use to size your bets, and

sqrt() is the "square root" function.

So here's how I figure my "Trip Bankroll." To get the SD/multiple trials (in this case it's 10), you take the square root of the "amount of trials less one". This is otherwise pronounced, "the square root of n minus one." In my case it's the SQROOT of ten minus one. (The answer is 3.) You then multiply this factor by your SD/trial ("block"). So, in my case it's 3 X 1225 or for the sake of conservatism and rounding--$3,700.

But that’s just your trip SD at the end of the trip, not your ROR for the whole trip. This is the same concept that Idiot Savant proposed in the December 1993 issue of Blackjack Forum. The problem is that the SD at the end of the trip is dependent upon lasting long enough to get that far. Schlesinger calls this the “premature bumping into the wall” scenario. You don't want to know your SD at the end of the trip, you want to know the probability of hitting a certain SD at any point during the trip.

To find the true Trip ROR you would need to perform a double-barrier check. In Blackjack Attack Schlesinger concluded that “the probability of being behind by at least a pre-specified amount at some time during a trip is a little more than double the probability of being behind by that same amount at exactly the end of that specified time.” Basically, there is a very big chance that you will lose $3,700 or more before the end of 10 blocks.

Also, I get different numbers for your SD/block. You said that your SD/100 hand block is $1225, so your SD/Hand is $122.50. If you play 10 blocks (1,000 hands) then your SD = Sqrt(1,000) * $122.50 = $3873.79. You don’t need to subtract 1 from the number of hands. By subtracting 1 from the number of blocks you are actually skewing your results by 100 hands.

Kelly betting simply means that you multiply your percent advantage by your bankroll (lifetime) to decide each and every bet you make.

Don’t forget about variance! You have to divide each product by the variance of each bet (usually around 1.33) to get the true Kelly bet for that TC. Otherwise you are overbetting.

Your initial bet does not always equal your total bet for the hand…At a halves TC of +4, I have about a 2% edge over the house for said hand. According to Half-Kelly, I should be betting $550 (55k X 0.02 / 2). However, in a DA2-DAS game with a TC=+4, it's extremely likely that I'll have a DD or split or both which can quickly escalate to "Double Kelly" (or double trouble because you don't only give back your edge with double Kelly, you may also increase the House Edge!)

That depends on whether you are using your initial bet advantage (IBA) or total bet advantage (TBA) to calculate your bets. Typically EV refers to your IBA which will take into account all doubles and splits. As long as you use your EV you don’t need to adjust the Kelly blackjack formula.

Well, sorry for such a long post.

Not at all! I’ve really enjoyed all of your posts so far and look forward to hearing more from you. You've offered a lot of great advice.

-Sonny-