[ QUOTE ]
I just read the FAQ post on sticky bonuses at bonuswhores.com (
link to article ), and I'm curious if anyone (Bruce? pzhon?) can show me the derivations of some of the forumulas....
[/ QUOTE ]
I haven't read all of the links yet, but I can answer the questions.
[ QUOTE ]
D=deposit
B=bonus
X=your target (you aim for $X or $0=bust)
BR=your whole gambling bankroll
First question is about your chance of success, which (for a fair game) is (B+D)/X. I can see that this formula is true when I plug in some different numbers (eg, if X = 2*(B+D), you just bet everything and clearly you have a .5 chance of winning). But what I cannot figure out how to prove is 1) the formula in the general case and 2) how to show the forumula does not depend on your bet sizes or betting strategy. I'd like to see how these two things are done.
[/ QUOTE ]
Since there is no house advantage, your expected ending amount is the same as your starting amount, B+D. If you end up with either 0 or X, you have X with probability (B+D)/X.
[ QUOTE ]
[ QUOTE ]
Since there is no house advantage, for any X, you can extract more EV out of the bonus. Let's say you've reached a value of X, and you set your goal to reach X+dX or bust trying. Simple calculation shows that for this bet:
EV=B/X * dX
SD=(X-B)/sqrt(X) * sqrt(dX)
[/ QUOTE ]
How were these two equations arrived at?
[/ QUOTE ]
You can compute the EV as above, but you start with X and your new target is X+dX, so you win with probability X/(X+dX). The improvement from ending up at X+dX rather than X is dX while the loss from ending up at 0 is X-B. You can compute the SD from that.
[ QUOTE ]
[ QUOTE ]
Using arguments similar to the ones justifying 300BB bankroll requirement in poker, my personal criterion for bankroll is: (I am conservative when it comes to bankroll)
BR = 4*Var/EV
I think this is equivalent to requiring ~500BB bankroll for limit hold'em.
[/ QUOTE ]
First, does this mean that with a BR of (4*Var/EV), you will have the same chance of busting out while playing sticky bonuses indefinitely as would a winning player (based on some reasonable winrate like 1BB/100 or 2BB/100) at limit holdem?
[/ QUOTE ]
Yes. The SD of limit Hold'em is roughly 15 BB/100.
[ QUOTE ]
Second, where does this formula come from?
[/ QUOTE ]
In some sense, this comes from the Central Limit Theorem.