I posted in the 2:1 promo thread about my plans to play video poker to satisfy wagering requirements on a large bonus. In my explanation, I referred to a percentage that I calculated like this. Let's take a 9/6 Jacks or Better $2 game, which has an $8000 royal. I know from playing progressives that this game would be breakeven if the royal paid $9850 or so. Another way of looking at this 'breakeven' point is to understand that the average player will lose $9850 before hitting the royal. Some players will hit it soon and have a profit afterwards, while others will lose more than that bankroll before ever hitting it. But, if you average them all together, you should get $9850, right? In the other thread, I figured a bankroll of $8000 meant that I have a ($8000/$9850) = 81% chance of hitting a royal before I go broke. As soon as I posted it, I realized it was a huge error. If that concept worked, a bankroll of $9850 would guarantee you'll hit the royal. My question: How do you translate a bankroll into a percentage chance of hitting the royal? I think this should be close: Without the royal, Jacks or Better returns 97.53%, which means your burn rate is 2.47% until you hit the royal. (I'm assuming straight flushes happen about as often as expected.) Therefore, on a game betting $10 per hand, the burn rate is 24.7c per hand. The $8000 bankroll should last about 32,388 hands at that rate. A royal is a one in 39724 shot, or p(RF)=0.00002517 on each hand. How likely are you to have no royals in 32,388 hands? That's (1-p(RF))^32388. p(Hitting no royal in 32388 hands) = 44.25% p(Hitting at least one royal in 32388 hands) = 55.75% So if I play my $8000 down to zero or until I hit a royal, I have a 55% shot of making it.
Another issue It occurs to me that there's another issue that reduces the percentage chance of hitting a royal with a given bankroll. That's the 'boundary problem', when you run out of funds. My estimate of how many hands the bankroll will last uses an average return, but it ignores the fact that you can't rebound from a negative total. I believe Don Schlesinger researched this issue for blackjack, and the same concepts should apply to VP. I'm sure the vpFree regulars could help quite a bit on this, and I may post over there.
