Does the gamblers fallacy not apply to Bayesian probability? Bayesian probability is an alternative probability theory that uses data from past outcomes to predict future outcomes. Do they have some work-around for the gamblers fallacy or do they just ignore it?
Or is Bayesian probability literally just the gamblers fallacy by another name?
 A: The predictive power of Bayesian inference only applies when the probability of an event is unknown, such as testing fairness of a possibly biased coin. It does not allow bypassing the gambler's fallacy when the probability is known or can be calculated, as is the case with most casino games.
A: In fact, a Bayesian approach is the exact opposite of the gambler's fallacy.
The gambler's fallacy supposes that if a certain event has happened more often than expected in a series of independent trials¹ - for example, a roulette wheel has been observed to mostly pick red numbers over the last day - it is likely to happen less often in the future, until it has evened out. A fallacious gambler will therefore bet on black in this instance, because a black is "due".
The Bayesian approach is that, supposing there was some initial uncertainty about the exact probability of the event, the fact that it has happened more often provides some evidence for what the actual probability is, and it will tend to revise our estimate for the probability upwards. So a Bayesian gambler will therefore bet on red, reasoning that the wheel may be slightly biased in favour of red.
This method of identifying small biases from observations of roulette tables was famously used to "break the bank" at Monte Carlo in the 19th century by Joseph Jagger.

¹ If the trials are not independent, e.g. cards dealt in blackjack, this may not be fallacious.
