Assumption: I always play red.
In our casino each customer can buy 6 discounted (golden) 5 EUR chips (30 EUR total) each day for 27 EUR. You can't change them back to money, so you have to play with them till all are lost, but what you win you get in normal chips which you can change to EUR. So you put a golden chip on red. If red comes you get a green 5 EUR chip and keep the golden chip. If black comes you lose the golden chip.
There are 37 numbers, 18 are red, 18 are black, and the 0. If you play on single chances (red/black/even/odd/1-18/19-36) and 0 comes you lose only half your chips. If you have an odd number of chips one chip is put on hold (on the edge of red) and you lose half of the remaining chips. On red, held chips turn back to normal and you can take them off the table (but I don't). On black, held chips go to the other edge of red to double hold. On red, double held chips go back to single hold. On black, double held chips are lost.
I made an iterative program which simply assumes I lose half on 0. The result is 97.3% return, so my win is 30 EUR * 0.973 - 27 = 2.19.
$x=100; #initial wager
$r=0; #return
for ($i=0;$i<40;$i++) {
$r+=$x*18/37;
$x=$x*18.5/37;
print "$r\n";
}
Is my simplified result correct? How close would you estimate it to be to the exact result?
If you feel like it, the exact expectation would be interesting, considering that double held chips can get lost. For this case I'd put all remaining golden chips on red, not just one or some, to avoid chips getting held. OTOH, what would be the result if I only played single chips?
I never play green chips. Once all golden chips are lost I cash out.
It's pretty easy to figure out the overall gain as the \$3 for discounted chips minus the loss due to the $0$ which is $3 - (30 - 30\cdot 18/18.5) = \$2.19$.
In theory this works, in practice for a single play cycle, a standard deviation of 3 means it can easily vary from 15/36.5 to 21/36.5 wins. For an overall average of $2.19 gain per day, it isn't going to break the casino or make you rich.
A retired casino floor manager once said, the only way to leave a casino with a small fortune is to arrive with a large fortune.