SCORE is a DERIVED measure....
LC,
There is a short answer and a rather long answer.
The short answer is for any advantage player, at a given time, will be betting with a given spread, say $B_min to $B_max, with intermediate amounts for counts in between. There are also the given rules, and it depends on whether they leave at certain negative counts - these will all affect the result.
But lets break things up: [assume they true count Hi-Lo in a 6-deck game]
1/ Divide everything by $B_min - so that a $20 to $240 spread (assuming play-all), starting to rise at TC=+2 (say $40) up to $240 at TC=+8 [a bad spread] becomes:
Bet 1 unit at +1 or less
Bet 2 units at +2
.
take your pick < +8
Bet 12 units at +8 or greater. [ok so far...right]
This is known as a 1-12 unit spread (or 1-12u for short).
2/ Say the EV per 100 rounds is about $25 (I said is a bad spread), and the SD per 100 rounds is about $750.
Firstly convert this to per round, remembering that the EV is divided by 100, but the SD is divided by 10 (ie VAR is divided by 100).
So EV/r = $0.25 and SD/r = $75.00.
Then doing the division gives: 'ev' = 0.0125u ; 'sd' = 3.75u .
The quantities 'ev' and 'sd' are the unit EV and unit SD per round. If you think the 'ev' looks small you are right, but since the bets range between 1-12u (mostly at the low end) the 'sd' is reasonable.
In general if B_min is just called $B ($20 in this case), and keeping things per round (the 100 round thing is just confusing) then we simply have:
EV = $B * ev
SD = $B * sd
VAR = $B * $B * (sd)^2
3/ Probability theory for large N, says that you have about one 33% chance of being 1 standard deviation below expectation after SQRT(N)*SD. Now this works for either EV(ev) or SD(sd) as the $B cancels out, so an unlucky player(33%) could be zero (or less!) after N hands if
0 = N * EV - SQRT(N)*SD or N*N*EV*EV = N*SD*SD .
Canceling the N and rearranging gives N = (SD/EV)^2 = VAR/(EV)^2 where we get the (famous?) value
N0 = (sd/ev)^2 = (SD/EV)^2
which is called the long run index.
4/ You will have to take my word on this, but Kelly Theory says that for a game using unit 1-M spread, with a given 'ev' and 'sd' has what is called an Equivalent Kelly Bankroll (ekb) given by
'ekb' = sd^2/ev = var/ev.
It is no coincidence that if we put back the $B, we get the dollar EKB
EKB = (SD^2/EV) providing the EV is per round, not 100.
So for the example above we get
N0 = [(3.75)/(0.0125)]^2 = 90000 rounds (lousy)
ekb = (3.75*3.75)/(0.0125) = 1125
and EKB = $20 * 1125 = $22500
(which means if this is your bankroll you have a risk of ruin of 13%. Not a good return on investment. Also notice (it drops out from the math) that
EKB/N0 = $0.25 = EV - not a coincidence!
Therefore your win rate per hand is simply your Equivalent Kelly Bankroll divided by your long run index.
Optimal Betting Theory says that simply by reducing your high bet down to TC=+4, you can halve N0, lower your EKB and increase your win rate at the same time.
But there is another trick - for a given 'ekb', you can choose your unit bet $B so that you can match your Kelly Bankroll. Now what happens if you optimise your betting spread (minimise N0), and adjust your $B such that your EKB is $10000???
You guessed it - your win rate EV/r = $10000/N0 !
Then if you wanted to have a win rate per 100 rounds you get
EV(100) = ($10000/N0)*100 which just happens to be the SCORE!!
I originally came up with N0 with my work on Optimal Spread Theory. Any N0 more than around 20000 is probably unplayable, unless you Wong which really can lower it, and good single deck games should be less than 10000.
Don felt that players relate better to win rate rather than long run chances of being ahead, so he invented SCORE - which is really just (1/N0) multiplied by $1,000,000. The main difficulty I have with it is just that players may just consider it a win rate, which can be calculated for any non-Optimal spread (that is to really 'KNOW' a game, you you need to know the ev and sd (or equivalently NO and ekb, not just N0 which you get from SCORE).
In other words, you should know the $B Don has used to convert the unit 'ekb' to $10000. He may well do this in his book, you need to check.
I believe QFIT has programs like CVCX to compute Optimal Betting Spreads - doing a Google is a good idea.
If all this has been a bit messy, you can always go to:
http://www.blackjackforumonline.com/content/TOClibrary.html#Bettingsystem
and find my two articles under 'Blackjack Betting for Card Counters'. The first 'How-To' article explains the stuff I said above, the second one is really only for the mathematically inclined who may have access to simulator data.
Cheers,
Dr. Brett Harris.