In last night's episode of Law& Order SVU, the villain forces Agent Benson to play Russian Roulette with him as his captive. He puts a bullet in one of the six chambers, and spins it shut. Then he says "I'll be a gentleman and go first."

At first I thought he was doing this to have an advantage, since he's least likely to find a bullet if he goes first. But then I got to thinking, is it actually better to go first? If you go second, you're freerolling for your life. Surely there's value in that. But the cost of that freeroll is that when he doesn't find a bullet 5/6 times, you have to avoid it 4/5 times.

There must be some math involved.

In russian roulette is the chamber re-spun every time, or do you just spin one and keep shooting from there? If it's just spun once, I don't think it matters. If you go first, you get chambers 1, 3 and 5. If you go 2nd you get chambers 2, 4 and 6. The position of the bullet is fixed after the spin and so you're essentially flipping a coin to see if it's in your half or the other half.

In retrospect I think they must respin every time or who would take the last shot? In that case you'd like to go 2nd.

Obviously if it is re-spun every time there is an advantage to going second. If it were some "normal game" there would be some possibility it could end in a tie (when both players shoot themselves). It seems that only the most "honorable" of captives would elect to spin one more time (to be fair) if the villain has already killed him/herself.

Why the tentative language? Of course it doesn't matter, just like in hold'em the 2nd player dealt has the same odds of getting an Ace as the first player dealt.

A person of honor. Or, I guess you'd need witnesses with guns so that neither player can renege. Otherwise the 1st player can just point the gun at the other and pull the trigger quickly 6 times and collect the monies.

I should have mentioned that it was not respun each time. I guess that was implied when I said it's 5/6 the first turn, then 4/5 on the next, etc.

(5/6) * (1/5) = 1/6

Edit: that basically shows why there's no positional advantage.

The full calculation would be this:
Pr(2nd player wins) = (1/6) + (5/6)(4/5)(1/4) + (5/6)(4/5)(3/4)(2/3)(1/2) = 1/2

The way we play is that we first spin the gun to determine who will go first. Then each player in turn inserts a single round, spins the chamber, and while it's still spinning, randomly relatches the chamber. This way no one has any advantage for the overall game, and everyone has the same 1/6 chance at the time he pulls the trigger. However, whoever gets to go first is at a disadvantage. If we were to only spin once, then each player would have the same chance no matter who goes first, but they wouldn't have the same chance when the trigger is pulled. If you play that way, then you should only have each player take a maximum of 2 turns, since if they each take 3, then sometimes the last guy would just be committing suicide, and that's not really Russian roulette. The thrill of the game is the anticipation of not knowing if the gun will go off each time you pull the trigger, and the exhilarating relief when it doesn't. That's really why we play.

I never knew you were such a badass! How much are you up lifetime from the game?

Russian roulette is a freeroll.

I find the many worlds interpretation really depressing, usually, but this makes it a little better.

I couldn't think of a way it would matter, but that doesn't mean there wasn't one.

Yes, but in hold'em if the first person receives a bullet you don't get to walk away a winner without getting dealt any cards.

This was sort of my first thought too. In Russian roulette if the first player dies then your odds of getting a bullet are now irrelevant . So your chance of dying is with the given that the first player does not die.
With or without respin, you always want to be second.

Someone should do a sim both ways.

Does anyone see a problem with this?

I mean aside from the fact that we can negate any gravitational bias by pointing the gun down. Here's an experimental result:


BTW, the guy on SVU let it come to a complete stop, and he probably could see where it landed.

And there are no ties, so Pr(1st player wins) = 1/2 also

Also, think again about hold'em. Pretend we remove 3 aces from the deck (or suppose we're only interested in the Ace of Spades). Are you telling me the 1st player dealt has a better chance of getting the Ace than the 2nd player?

By your argument, "once the first player gets the Ace, the 2nd player's odds are irrelevant". Clearly a fallacy.

Gotta remember what OP pointed out -- if the 1st player misses, the 2nd player's conditional probability is increased. Most of the time that increase will happen, and so that cancels out with the few times his chances are reduced to 0.

Scientists Claim That Quantum Theory Proves Consciousness Moves To Another Universe At Death

The 3rd most important scientist alive today believes this, and he's supported by distinguished physicist Sir Roger Penrose and others.

Penrose has kind of clearly lost the plot. I think he has some good points but he has more hand waving and wishing than anything else. It's OK though. /notaphysicist

I do think his strongest point is that there are now big parts of physics that (currently at least) can't be used to predict anything, and as such, aren't... real.

I wonder if theoretical physicists are more likely to go insane than the rest of the population.

With no respins, going first is clearly better.

If handed the gun after five blank pulls, you know the next pull results in a bullet. So you shoot the other guy.

After four blank pulls, if you play fairly, you're certain to die either with the next pull or by handing the loaded gun to the other guy who shoots you. So you shoot him, 50% of the time he dies, 50% of the time the game ends on a penalty to you.

After three blank pulls, if you play fairly, there's 2/3 chance of dying, and no chance of killing. So you shoot the other guy, 1/3 chance of killing him, 2/3 chance of game ending on a penalty.

After two blank pulls, if you pay fairly, there's 1/2 chance of dying and no chance of killing, so you shoot the other guy. 1/4 chance of killing him, 3/4 chance of game ending on a penalty.

After one blank pull, if you play fairly, there's 2/5 chance of dying and no chance of killing, so you shoot the other guy, 1/5 chance of killing him, 4/5 chance of game ending on a penalty.

When handed the freshly spun gun, if you play fairly, there's 1/3 chance of dying and no chance of killing, so you just shoot the other guy. You keep pulling the trigger until you succeed. 100% chance of winning, with no one to complain about the penalty. So going first is better.

I get that the math is 50/50 but there are still two advantages to being second. 1/6 you won't have to pull the trigger at all and that fact has value.

And, in a game with no respin then 1/6 you will be holding the gun after 5 clicks with all uncertainty removed and at the point will uniquely have the choice of certain suicide or not. I would suggest that a fair game would have to call a draw after 5 clicks because otherwise one player or the other is always going to murder the other at that point. And with such a draw rule, 2nd has the probability advantage in a no respin game. Yes this is changing the rules but with no respin, I see no rational alternative.

If I go 5th and dry fire, then I just point the gun at your head and fire the last round for you.

The math is that if 1st person dies before you the game ends (1/6 of the time) before you even spin. Is it better to have 5/6 chance of surviving or (1/6+4/5 chances of surviving). BOOM.

It's not 1/6 + 4/5, it's 1/6 + 5/6*4/5 = 5/6. You have the same 5/6 chance of surviving the first round regardless of whether you go first. Obviously, since there's an equal chance of a round being in chamber 1 or 2.

That's if the cylinder is only spun once. If each player spins, then the first player has a 5/6 chance of surviving the first round, and the second player has a 1/6 + 5/6*5/6 = 31/36 chance which is better.

I was just about to say I don't understand, but now I understand. LOL my bad. I take back the boom. Still doesn't seem to make sense that the first spinner has to spin, but the second might not have to spin if the first dies?! How is that same chances of surviving?????

but in holdem if the first player is dealt an ace, it lowers the second players chances of being dealt an ace too.

My gut feeling tells me I would rather be the 2nd person pulling the trigger because I might not have to even pull it if the first dies. I think my survival instinct is trying to make me reneg on me understanding the math now. I'm gonna go think about something else for alil while.

One last thought... Is this anything like headsup nlh where statistically speaking before the cards are dealt there is a 50% chance either the button or the bb will have the best hand at the end of the hand if it is showed down, BUT the button has the statistical advantage because of post flop play/being 2nd to act on all post flop streets(getting to pull the trigger 2nd)????

although the button has to act first preflop.... back to where I started I guess.