The question you opened
Nine reds in a row
A European wheel with 37 pockets has landed on red nine times running.
What is the chance the next spin is red, in percent?
48.6 percent
The wheel holds 18 red pockets out of 37, which is 48.6 percent. The pockets do not rearrange themselves because of the last nine spins.
A European wheel has 18 red pockets, 18 black, and one green zero, so 37 in total. The chance of red on any spin is 18 divided by 37, about 48.6 percent, on this spin and on every other one. Nine reds in a row change nothing, because the wheel has no memory and no mechanism by which past results could influence the next drop. The feeling that black is due is called the gambler's fallacy, and it comes from a real fact applied to the wrong object.
Over a very long run the proportion of reds does approach 48.6 percent. That happens by dilution rather than by correction: the nine extra reds are never cancelled out, they simply become a smaller and smaller share of a growing total. The famous instance was at Monte Carlo in 1913, when a wheel came up black 26 times in a row and players lost large sums betting on red because it had to be due.
The chance of 26 in a row is roughly one in 137 million, which is exactly the sort of number that appears somewhere once enough wheels spin for long enough. There is a mirror error worth naming. The hot hand fallacy assumes the streak will continue because the wheel is running red. Both errors read a pattern into a device that produces patterns by construction, and they lead to opposite bets on identical evidence.
One honest caveat. If a wheel really produced a long enough streak, the correct conclusion is not that black is due, and not that red is hot. It is that the wheel may be biased, which is what Joseph Jagger exploited at Monte Carlo in 1873 by recording thousands of spins. That inference needs data on the wheel, and nine spins is not data.
Technique: Each spin starts over