Martingale System for European Roulette

Let's say I am playing a martingale system with a European Roulette (1 zero), choosing either black or red. My chances of winning at each spin are 18/37 and my chances of losing at each spin are 19/37. Most casinos only let you double 6 times (example: min $5, max $250, so you can do 5-10-20-40-80-160).
If I play in a manner that I pick the same color and double on that same color until I win and then switch over to the other color and double until I win and back and forth. Therefore, I know that the chance of having 6 of the same colors in a row (and therefore losing $325 (5+10+20+40+80+160) is (19/37)^6 = 1,86%

So here is my question: After how many spins should I just take my winnings and leave? I mean that playing a martingale and winning 5 cycles of martingale original bet (in this case $5 to start with and leaving the casino with $25) carries less chance of having 6 of the same color in a row compared to playing 50 cycles of wins which would bring in 50X$5 but much more chances of getting 6 of the same colors in a row. So after how many spins is it statistically best to stop playing and take smaller winnings home?

Quote: fabio1957

So here is my question:
After how many spins should I just take my winnings and leave?

The probability of one win = 98.1664%
It does not matter how many spins you see each 10 minutes or 10 years.
Each spin is independent of past spins.
You need to understand that fully to move forward.

lets say you want to double your $315 starting bankroll (6 step Marty)

You MUST win 63 times in a row or you will lose $315
(since each win at any step increases your bankroll by just $5)
and it does not matter if you win just $5 every 2 weeks.

so the probability to double before the first Big loss then is
98.1664%^63 = 31.163960%

Less than a 1 in 3 chance.

That is all you want?
Not me

You can get close to 48% or higher with other games or bets just flat betting.

You really are gambling after the first win
You have about a 50/50 shot of making it past 37 wins in a row (37*$5)
not even enough to cover your first $315 loss
=================================
Of course, after your first loss you will probably still have some bankroll left
and you might just bet it all
to try to get back to your starting $315 while on tilt.
Not a bad idea.

IF you just want to keep trying your Marty with less than $315
you have a (by simulating 100,000 times)
[1] 0.4286605
> p-error
[1] 0.4234978
> p+error
[1] 0.4338231
probability of an overall double to your bankroll.
So all the fun would not be gone after the first big loss, but it just might be

Good Luck and have fun