Well two things:
First, you cannot use binomial distribution for events which rely on each other (lose 3 in a row) unless you calculate the odds of losing 3 in a row and use that as a single failure event. Problem is a single failure event "uses up" rolls. Ie: if you are rolling 20 times, but have 3 failures, 6 of the 20 are "used" up as part of the failure run. Hence my use of "runs" instead of rolls.
Right. I don't think anyone is counting rolls. We're counting "martingale sequences" - start at your base, play until you win or bust, that's 1 sequence.
Second: if you have a 12.5% chance of failure (your 1/8th example above) and you make 6 attempts, what is your chance of failing? It's not 1in8 / 6. The ONLY way to accurately gauge it is the BD(tired of typing that).
I don't think that's true.
If I make 6 attempts and each one has a 1/8 probability of failing, the probability that at least 1 of them fails is 1 minus the probability that none of them fail, ie.:
1 - (1 - 1/8)^6
= 1 - 0.875^6
= 1 - 0.4488
= 0.5512
No BD in sight. No factorial, etc.
Thinking a .02% chance (1 in 2000 chance) of failure, and you make 100 runs, and you'll only have a 1 in 200 chances of losing will make you go broke when you lose a hell of a lot more often.
First off, you made a couple of mistakes:
1) I think you meant ".05% chance" (since that is 1 in 2000 : 100 / 0.02 = 5000 and 100 / 0.05 = 2000)
2) Also, did you mean "1 in 20" not "1 in 200" (since 2000/100 = 20)?
Anyway, the truth is that when the number of runs you're making (100) is significantly lower than the odds against failure happening (2000) then the naive guesstimate ("2000/100 = 20, so it's 1 in 20") is pretty accurate, and a little conservative. It tells you your chance of failure is HIGHER than it really is.
In fact when you make 100 runs and each has a 1 in 2000 chance of failure, your overall chance of failure is lower than 1 in 20. It's more like 1 in 20.5 - to be precise, it's 1 in:
1 / (1 - (1 - 1/2000)^100)
= 20.49916711818098
Like you said, if you make 2000 runs with a 1 in 2000 chance of failure per run, the overall chance of failure isn't 1 in 1. The actual chance of failure is lower than the naive runs/odds calculation would lead you to believe. In this case it's 1 in 1.58:
1 / (1 - (1 - 1/2000)^2000)
1.58174652400438
