Replies
Overkill,
Here are my replies:
You wrote:
2) I thought there was a handy standard deviation formula that gave you an estimate of how likely it is to, say, be ahead (but not by luck) by, say, 27 units after 400 hands?
Let's take this one first, as the answer to (1) can be found with these same techniques. If you know the EV (expected value) and the SD (standard deviation) of the game you're playing, you can use these values to determine the likelihood of being up by 27 units in 400 hands. By the way, the EV & SD will depend on the house rules (# of decks, S17/H17, DAS/NoDAS, etc.), the penetration, the count system you use (including the indexes), the actual "bet schedule" you use, and the total number of players in the game. If you use a commercial software package, like CVCX, you can input the parameters and it'll give you the EV & SD.
Let's just pick some fairly typical values to illustrate how to answer your question: say the EV/hand is 0.02 units, and the SD/hand is 1.15 units. NOTE: this SD/hand is approximately correct for FLAT BETTING: if you spread your bets (as card counters generally do), your SD/hand will be SIGNIFICANTLY larger.
Now in 400 hands, your expected win (EW) is
EW = (EV/hand)*(# of hands) = (0.02 units/hand)*(400 hands) = 8 units.
Now the expected deviation (ED) is a bit trickier: it is NOT equal to the (SD/hand)*(number of hands). Instead, it varies as the square root of the number of hands played. Therefore:
ED = (SD/hand)*(# of hands)^(0.5) = (1.15 units/hand)*(400 hands)^(0.5) = 1.15*20 = 23 units.
Now one property of the ED is that, approxiately 2/3rds of the time, your actual result will be within ONE ED of your EW. Thus, if you repeated this experiment numerous times, approximately 2 out of every 3 experiments will result in an actual win of between 8-23 = -15 units and 8+23 = +31 units. Of the remaining 1/3rd of the results, half (so 1/6th of the total) will be more than +31, and the other half (so 1/6th of the total) will be less than (more negative than) -15 units. Oh, the "2/3rds" and "1/6th" are approximations: the actual fractions are 0.682689... and 0.158655... You can get these values in Excel by typing the formulas =NORMSDIST(-1) (to get the 0.158...) and =1-2*NORMSDIST(-1) (to get the 0.682...).
Now to answer your question: the probability of your being ahead 27 units in 400 rounds can also be found in Excel. Since your EW is 8 units and your ED is 23 units, entering the formula =NORMSDIST((8-27)/23) will give us the result of 0.204377... This means you have about a 20% chance of being ahead by 27 OR MORE units after 400 hands.
Of course, these results all depend on the particular values we use for the EV and SD: if we select different values, we'll get different results.
Now we can get back to question (1):
You wrote:
My strategy was ahead $50,000 (betting $100/hand) after about 298,000 computer-simulated hands. So, if I were at the Bellagio and betting $50,000 per hand, I would have been ahead about $50,000,000! I don't think I'm in error yet, correct?
OK, but after a total of about 623,000 hands and still betting $100/hand, I'm down about $136,000.
1) Is my above sceanario (whereby one is ahead after about 1/3 of a million hands but behind after 3/5 of a million hands) fairly common?
Here again, the answer will depend on the EV & SD. To illustrate, though, let's use the same SD as before, but a lower EV of only 0.01 units/hand. First, we want to know the probability of being ahead by 500 units ($50,000 with $100 units) after 298,000 hands. The EW and ED are as follows:
EW = (0.01 units/hand)*(298,000 hands) = 2980 units
ED = (1.15 units/hand)*(298,000 hands)^(0.5) = 627.8... units.
Since you were up "only" 500 units, you underperformed ;-) The probability of being up 500 units OR MORE after 298,000 hands is then
Probability =NORMSDIST((2980-500)/627.8) = 0.999960997
Thus, it's a near certainty: the probability of being ahead by LESS THAN 500 units (including being behind) is 1 - 0.999960997 = 0.000039..., or about 1 chance in 25639.
You can now work out the Probability of being BEHIND by 1,316 units after 623,000 hands... hint: it'll be REALLY small!
You wrote:
3) Do I need at least one billion hands to be 99% confident walking into a casino?
No... usually a 400-million-round simulation is sufficient to tell us the results.
You wrote:
4) Even if the House still has the advantage (say,.17%), can't a player walk away with a hefty sum if betting heavy enough (with relatively small risk) for a very short time (~12,000 hands, or whatever). Is there a formula I can use to calculate risk given house advantage?
We can apply the same ideas to answer this question. Let's again assume a SD/hand of 1.15, but now the EV is negative, since the house has the edge: EV = -0.0017 units/hand (for a H.E. of 0.17%).
Now we want to know the probabilty of being even (win of $0) or ahead after 12,000 hands.
EW = (-0.0017)*(12,000) = -20.4 units
ED = 1.15*sqrt(12,000) = 126 units (approximately)
Probability =NORMSDIST((-20.4-0)/126) = 0.435678377...
So we see that you have a better than 2 in 5 chance (better than 40%) of being ahead after 12,000 hands even if the house has the edge. Of course, that ALSO means that you have nearly a 60% chance of being BEHIND after 12,000 hands.
Finally, you wrote:
5) Does one really need at least a .0000000000001% edge to be able to walk away a winner? In other words, what is a reasonable disadvantage to accept and still expect to win (say, $4,000,000) by betting big in the very short run?
WOW! Find a craps table with high limits. Plunk $4,000,000 down on the Don't Pass line (H.E. of 1.35% if I recall correctly) and root against the shooter ;-) You have a 49.325% chance of winning $4 million... and all the comps you'll EVER need!
Hope this helps!
Dog Hand
