Do you believe Bayes' Theorem is correct? It's not controversial in probability, and is easy to prove.
I believe Bayes' theorem correct, even without proof. Bayes' theorem describes exactly the way we perceive reality, or how people think in everyday life (and, most of the time, they're correct).
The police suspects and arrests people almost exclusively based on Bayes' theorem. Your wife is killed, most wifes are killed by their spouse. Therefore you're guilty. You are (insert ethnic group). Most thefts are done by (said ethnic group). You happen to be close to the crime scene, so you're the thief.
Funnily, more often than not, it turns out they're exactly right with this dumb approach, too. So, yeah, Bayes works.
However, applying Bayes' theorem (or any such thing as probability, for that matter) when it isn't applicable is wrong. After having shown that five chambers are empty and knowing that you've put one bullet in (and you haven't taken it out, or spun the drum again, or otherwise cheated), there is only one possible outcome in a universe where guns and bullets are physical objects.
(long explanation)
There is no fallacy to it, even though your explanation sounds perfectly reasonable. It doesn't matter whether or not you've survived the previous 5 rounds, since if you haven't survived (or the other guy hasn't) then you don't meet the experiment's conditions. It's pointless to consider whether or not that happened. Also, you cannot change past events based on observations on present (or less remote) events.
The initial assumption is that you have arrived at the last chamber, and there is a 50% chance that the referee has put in a bullet.
If there is a bullet, the bullet must be in this last chamber (there is no other way!), and the referee made his decision whether or not to put one in before anyone touched a trigger. The 50% chance that there is a bullet in one chamber doesn't change after the referee has tossed his coin, merely because nobody has died during the first five rounds. The only thing that changes during the game is the number of chambers that are left (and people dying, but that is outside the frame conditions of the experiment).
Our universe works in a way that you can best describe with memory_order_seq_cst. Things (even unrelated ones) happen in a sequentially consistent way (at least at non-relativistic speeds and with real objects and without gremlins reloading your gun). The referee's decision happenes first, the state of the gun is globally visible before any player touches the gun. The gun's "loaded" state is defined a priori and doesn't change afterwards, regardless of whether you've survived the first rounds or not.
The fallacy is in thinking that your survival of the first 5 rounds can change the past.
