@Todd ,Mathman & Michael:
Yes your results are matching with mine. In fact I did this on Pari/GP & Mathematica and with about 10^9 run I got probability very close to 30% for B with average 19.98 rolls.
As Mathman rightly said I also thought it should take more rolls based on average advance. That's why I suspected my result and wanted to verify. With such clean answer (approaching 30%) I am sure analytical solution must exist, but not sure how to proceed.
Here is the best example http://answers.yahoo.com/question/index?...
Vasek: Thanks for your programmatic implementation. (I made typo in my earlier comment where I noted 19.98 as avg rolls it was 18.98. I think noise in my answer was different due to excessive number of trials)
I doubted my answer more because , I could not believe that on an average Player B would require only additional 2 steps.