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What are LN and WR in the above equation? SD is standard deviation?
Avoiding ROR was bad wording, but what I was looking for was an equation that Ic ould plug in different RoR% and see the changes.
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LN is the natural logarithm, WR is the win rate, and SD is the standard deviation. Here is an easier equation you can use which is also exact:
ror = (0.451/0.548)^B
where B is the bankroll size in bets. For example, if you want a ror of 1%, B = 23.64 bets. So if your bankroll is $100,000, each bet can be 100,000/23.64 = $4230.
0.548 is your probability of winning a bet since you always win 99/1000 times that he picks 1-99, and you win half the time that he picks 100-999 unless you tie, so your probability of winning is 99/1000 + 900/1000 * (.5 - 1/901) = 0.548. His probability of winning is 0.451
Solving for B gives:
B = ln(ror) / ln(.451/.548)
where ln is the natural logarithm (or you can use base 10 logs here too). For a derivation, see the first part of my derivation of the full ror formula in
this post.