N0 and Probability of Being Ahead

assume_R

Well-Known Member
#1
Is there an equation in which you can calculate the probability of being ahead after X hands?

I assume that N0 is actually X for a certain probability (something high like 95% I'd guess).
 

Sucker

Well-Known Member
#2
There is a section in Peter Griffins' "Theory of Blackjack" that gives the equation you're looking for. My copy is 2000 miles away from me at this time so I can't tell you which page it's on.
 

SleightOfHand

Well-Known Member
#3
assume_R said:
Is there an equation in which you can calculate the probability of being ahead after X hands?

I assume that N0 is actually X for a certain probability (something high like 95% I'd guess).
It is a simple statistic problem. If you know the standard deviation and win rate per hand...

mean = X * win rate
total standard deviation = SQRT(X) * standard deviation

Now, the probability of being ahead would be the probability of being at 0 or greater. To find this, you need to know how many standard deviations from mean this would be (z-score). This would be...

z-score = (0-Mean)/total standard deviation

Then, using a statistical calculator or the internet, find the cumulative probability of being at that z-score or higher.

Rules of thumb:
z-score of -1 would have a probability of ~84%
z-score of -2 would have a probability of ~97.5%
z-score of -3 would have a probability of ~99.85%

Hopefully I didnt screw something up.

PS: Since N0 is defined as when the mean = SD, this is equal to a z-score of -1 aka 84% chance of being ahead after N0 rounds.
 

blackjack avenger

Well-Known Member
#4
N0 Flat Line

the N0 for flat bets is
1,4,9
for 1sd, 2sd, 3sd

the N0 for Kelly continuous resizing is (4*N0)
4,16,36

the N0 for half Kelly continuous resizing is I believe? (1.87*N0)
1.87, 7.48, 16.83

If you resize your bets at all based on bankroll movemnet your N0 will be closer to the continuous numbers.

:joker::whip:
 

assume_R

Well-Known Member
#5
Thank you, Sleight, this makes a lot of sense. It also assumes that card counting results in a normal, or Gaussian, distribution, which I believe it does. Perhaps other games (such as progressive slots or video poker or something else) don't have a normal distribution and the equations would be more complicated, but that's another beast entirely.

Avenger, are you saying that when you continuously resize your bets based on your bankroll, it takes longer (higher N0) to be ahead??? Not sure I understand why that's the case or perhaps I misinterpreted your post.
 

blackjack avenger

Well-Known Member
#6
Long and Strong

assume_R said:
Avenger, are you saying that when you continuously resize your bets based on your bankroll, it takes longer (higher N0) to be ahead??? Not sure I understand why that's the case or perhaps I misinterpreted your post.
Yep,
When you are resizing and lose you cut your bet size, then it takes more wins in order to get back to even, which raises N0.

If you resize about every 25% of bankroll movement your N0 is about 10% less then the continous resizing N0 numbers.

Being willing to resize down on losses; which raises N0, is the cost of insurance against going broke.
 

assume_R

Well-Known Member
#7
Okay, yeah I understand the importance of bet resizing when your bankroll drops, but on the other hand aren't you also resizing up when you bankroll increases? I guess I just thought it would even out to keep N0 about the same.
 

blackjack avenger

Well-Known Member
#8
Long is Good

assume_R said:
Okay, yeah I understand the importance of bet resizing when your bankroll drops, but on the other hand aren't you also resizing up when you bankroll increases? I guess I just thought it would even out to keep N0 about the same.
Yes, you resize your bets up on wins and down on losses.

Perhaps a comparison with the same bank.

kelly fixed:
13.53% ror
$50 an hr ev (example)
3sd N0 = 9

kelly resized:
0% ror theoretical
3sd N0 = 36
$25 an hr ev (example)
Obviously an initial hourly ev because eventually you will far outpace fixed betting.

If you resize fairly frequently your numbers are closer to resized.:joker::whip:
 
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