Do you believe Bayes' Theorem is correct? It's not controversial in probability, and is easy to prove. The gun not firing in the first 5 shots is evidence against there being a bullet having been loaded in the gun, that accounts for something?

If the gun has 1,000,000 barrels and you fire it 999,999 times and there is no bullet what is the chance the next shot has a bullet? The same? What happens as you take the limit to infinity?

My number of hairs question was that the probability is 1 by the way, even if we assume max number of hairs on head is 10 million (average under 200 thousand), since there are more women than that if we give them a piece of paper with their number written on it and ask them to file them we find at least 1 file with more than one woman, this is the pigeon hole principle.

Together with the 50% chance of a bullet being there at all, the 100% chance of having it in your chamber if there is one gives you a 50% overall chance. No need for complicated proofs or experiments.

No, that is a fallacy, and one the paradox is designed to point out. Think about it, if the gun is empty then there is a much higher likelihood that both of you will survive the first five chambers (read: if you played many such games, where you and your opponent lived to the last chamber, then you will see the gun is empty more often than it is loaded - that is kind of the definition of probability). And so conversely, if you survived the first five chambers, then there is a higher probability the gun is empty than it is loaded. That probability turns out to be 6/7 by the proofs given above (so you die 1 out of 7 times) but that argument is enough to "intuitively" show it can't be 50%.

If you want, you can try out the problem "in real life" (with appropriate safety measures taken, please) and you will see that if you follow the problem statement carefully, the answer will come out at 1/7, not 1/2 like you say. It's not the mathematics that are wrong, it's your understanding of what the problem is saying and when it applies which is. That's why it's called a paradox, it's supposed to pick your brains. Don't take it personally, you're not the first (and won't be the last) to get all confused over things like this.

Every time the trigger is pulled and the bullet is not there the hypothesis that there is no bullet should gain some weight. Bayes's theorem is not an attempt at intimidation: It's just the formula that allows you to quantify this effect. And really, this is a great opportunity to learn about it, because if you can solve this problem by an argument like Geoffrey's, you probably don't need to even remember Bayes's formula: You can deduce it when you need it.

A friend of mine proposed a variation that might help some people think about this problem. Imagine we modify the game so that after pulling the trigger we spin the barrel, then pass gun to the other player. So instead of sampling the chambers sequentially, we pick a random chamber every time. The first time someone pulls the trigger, the probability of it going off is 1/12. Let's say it does not go off. Is it 1/12 for the next attempt? We actually do this 1,000,000 times and we haven't found a bullet yet. Do you still think the probability of it firing is 1/12 at the 1,000,001st attempt? Shouldn't you start suspecting at some point that it is more likely that there is no bullet, or otherwise you probably would have found it by now?

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.

You don't have to shoot anyone in the head to test it. All you need to do is fire a gun 5 times without it going off. Alvaro even did a simulation. You can try it yourself with playing cards anyway.

As Alvaro and myself pointed out if you have a large number of trials it gets more obvious.

Applying Bayes' Theorem to probability isn't controversial, and the proof is easy. Applying it to statistics as a measure for belief in something is controversial, we're not talking about that though?

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).

This is where your misunderstanding lies. There is a 50% chance that the referee has put a bullet in this gun. Not in this chamber. You don't know which gun you have. You might have the empty gun.

Think of it like this: You have a total of 12 available barrels. 1 barrel has the bullet.

Initial chance: 1/12

second chance 1/11

third chance 1/10

4th chance 1/9

5th chance 1/8

6th chance: 1/7.

It doesn't matter what random subset of barrels you take the first 5 samples from. If there wasn't a bullet yet, it has to be in the next 7 barrels.

That's a good explanation! Probs better if the other gun is in an alternate universe/parallel dimension though ;)

I tried really hard to understand Alvaro's example, but couldn't get my head around how it would be correct. I've never been good with probabilities, and I guess that illustrates the point he was trying to make about how people's intuition can be very wrong.

That said, WOW! WarAmp comes through with an explanation that makes perfect sense to me. You start at 1/12, and every click increases the probability a little bit for the next attempt. Nice one!