There's a famous problem in Probability called Gambler's Ruin, and it's quite interesting. Cottoneyedjoe had a recent post called "Are you smarter than a Finance major?" that broadly ties to this problem.
Suppose you start with "i" units (dollars, euros, crypto-currency, or whatever...) and your goal is to achieve "N" units, where N>i. Next you gamble a single unit at a time with a probability "p" of winning a single unit and "q" (=1-p) of losing the unit. You continue this until you either go broke (lose your "i" starting capital) or you achieve your goal of N units (at which point you stop betting).
What's the probability of achieving the goal (N units)? Here's the formula:
P = [1-(q/p)^i] / [1-(q/p)^N] for p≠q
and
P = i/N for p=q
Let's denote Q as the probability of ruin. Then since that is the only other outcome given the way we defined the problem, Q=1-P
Consider now that, perhaps, you don't want to contemplate losing your entire capital, so you set a lower stop loss limit. For the sake of argument, let's say we start with 100 units, and we stop when 30% of our capital is lost. Conversely, on the winning side, we stop when we've achieved a 10% gain. In context of the Gambler's Ruin problem, for this example, this is equivalent to i=30 and N=40 (i.e. we have 70 set aside that will not be touched in any circumstance).
Let's take the cases of p1=0.4995 and p2=0.5005, a 0.1% advantage for the House, and a 0.1% advantage for the player.
Then, if I did the arithmetic right,
P1 = 0.7425 and P2 = 0.7574
Because you set low expectations and have a high risk tolerance, you have nearly a 3 in 4 chance of achieving the goal even when the House has a built-in advantage! Great, should you give up your day job and become a professional gambler, just repeating this over and over again, and become rich?
Not so fast! The problem here is the Expected Value (EV) of the outcome. I've talked about EV in previous blogs. The "give up your day job" strategy only works if the ratio of the EV for each experiment over capital at risk is >1. Let's calculate that for the P1 case:
EV = P1*win-return + Q1*lose-return = (0.7425)(+10) + (1-0.7425)(-30) = 7.425 - 7.725 = -0.3
Since we risked 30 units, the normalized EV/unit-at-risk is -0.3/30 = -0.01. So if we continue to repeat, we'll lose about 1 unit per 100 units at risk on average.
Yep, reality sucks!