I am not sure if this part can be expressed mathematically, however I am also curious to know the chances of a gambler BOTH having a sequence, server/client seed combination, bankroll/starting bet combination that will allow him to 4x his money AND to stop gambling before busting. For example, a gambler who wants to 10x his initial bankroll who has a server/client seed combination (ect.) that would allow him to 6x his initial bankroll will end up busting, however a gambler who is in the same situation except whose goal is to 4x his initial bankroll will end up multiplying his initial bankroll by 4x even though he could have potentially multiplied his bankroll by 6x.
Well it's just statistics. If you have the persons "play strategy", the probability calculations are very easy. The problem is you can't reliable figure out someones "play strategy" just based on their bets. For example, you you have things like causation (e.g. maybe he wanted to double his money, but since things were going well so fast he decided to go more aggressive). Or maybe he withdrew as he just won an amazing streak, and was worried his luck would run out. Or maybe the play strategy was to stop after a certain time rather than outcome, etc.
There's also a lot of hidden variables, like a users "true bankroll". It's pretty common (especially in loss-chasing players) to only deposit a small amount of money, and then only if they lose they deposit more, and more, to chase that loss. But the casino can only see what they did deposit, not how much they were prepared to do so.
So I like to simplify it, and approximate the probability of someone getting X times their money is a little under 1 in X. It obviously will vary a little depending on their play strategy. e.g. if they do a single bet, they'll pay an expected the house edge. Various betting sequences (like a reverse-martingale) can even lower the expected player loss. Normal betting (Turning the money over and over) will increase the expected loss.