scepticus,

It's not correct to ask when a system will "meet its demise" because it may do that right from the first spin. The law of large numbers tells you that the probability will converge to its "true" value after a certain number of spins, with some degree of certainty (which you can make as high as you like). The standard C.I. (confidence interval) is 95%, which applied to your 8/1 bet gives 214,268 spins. If you increase the C.I. to 99% you will need 5 times as many spins.

When betting on more numbers, the number of spins needed is much less before convergence happens. For example, a 2 dozen bet will converge within about 14,000 spins, again with C.I. = 95%.

And these are conservative estimates. If you wrote a computer simulation you would find that the probability would converge MORE than 95% of the time.