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Probability – life or death question

May 10, 2012
Barrel_adjacent_bullets

Here’s a nice little problem:

You’re playing a game of Russian roulette with a six shooter. You know there are two bullets in adjacent chambers and the other four are empty.

Your opponent has just spun and shot at his own head. He survived and now it’s your turn.

Do you:

1) Re-spin and shoot

2) Just shoot

——————–
How different is it if you know the bullets are not adjacent?

How different is it if you don’t know whether the bullets are adjacent or not?

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15 Comments
  1. Anonymous permalink

    Once again (if you’ve had read my response to the Google Interview Question post, you’ll understand what I mean), in order to hide my response from anyone else who is considering their answer to this, my response is encrypted. Go to http://rot13.com/ to decrypt it (if you’re interested in reading it, that is).Lbh fubhyq er-fcva. Gur snpg gur bgure cynlre unf fgehpx na rzcgl punzore erqhprf lbhe punaprf bs fgevxvat na rzcgl punzore ol 1. Vs lbh qb abg er-fcva, lbh jvyy unir n 3/5 (0.6) punapr bs fgevxvat na rzcgl punzore naq 2/5 (0.4) punapr bs fgevxvat na ybnqrq punzore. Jvgu n er-fcva, lbh vapernfr lbhe bqqf bs fgevxvat na rzcgl punzore sebz 3/5 (0.6) gb 4/6 (0.666 erpheevat).Xabjvat jurgure be abg gur ohyyrgf ner nqwnprag be abg unf ab vzcnpg ba lbhe punaprf jura gur bgure cynlre unf fgehpx na rzcgl punzore. Nf lbh bayl xabj gur ynfg punzore gb or fgehpx jnf rzcgl, lbh unir ab vqrn jurgure be abg gur arkg punzore pbagnvaf n ohyyrg.Vs gur bgure cynlre unq fgehpx n punzore pbagnvavat n ohyyrg naq sbe jungrire ernfba lbh qrpvqrq gb cynl ba, vs gur ohyyrgf ner nqwnprag, lbh jbhyq unir n 1/2 (0.5) punapr gung gur arkg punzore jbhyq rvgure pbagnva n ohyyrg be abg (rvgure gur arkg punzore be gur cerivbhf punzore jbhyq unir gb pbagnva gur bgure ohyyrg vs gurl ner nqwnprag). Va guvf vafgnapr, lbh jbhyq or orggre bss er-fcvavat, nf guvf jbhyq punatr lbhe punapr bs fgevxvat na rzcgl punzore sebz 1/2 (0.5) gb 5/6 (0.833 erpheevat).Vs lbh qvqa’g xabj vs gur ohyyrgf ner nqwnprag be abg, lbh jbhyq unir n 4/5 (0.8) punapr bs fgevxvat na rzcgl punzore. Ntnva, lbh jbhyq or orggre bss er-fcvaavat nf guvf jbhyq vapernfr lbhe punaprf bs trggvat na rzcgl punzore gb 5/6 (0.833 erpheevat).

  2. Anonymous permalink

    <html><head></head><body bgcolor="#FFFFFF"><div>Thanks. I’ve chosen not to encode this as I think the discussion is more interesting than the answer!&nbsp;</div><div>In the first scenario, once the other player has shot and revealed a blank, I think this means there aren’t 5 possible chambers left for me.&nbsp;<br><br></div></body></html>

  3. @srcav permalink

    If the bullets are adjacent, the probability of dying is 1/4 if your opponent has shot a blank. if you respin then you have 1/3 probability of dying. To respin would be foolish.If they’re not adjacent then you would def want to respin as P(dying) is 1/2 without a spin but 1/3 with a spin.

  4. @srcav permalink

    The reason it is 1/4 is the two chambers immediately after bullets (a bullet and an empty) cant possibly be the next up, as you would have needed a bullet. This leaves you 4 possible chambers you could be on, one of these 4 contains a bullet. Thus less than a respin which has 2 bullets in 6 chambers leaving 1/3 chance of dying.If they are not adjacent, but the chambers you rule out are empty, leaving 2 bullets left from 4 possible chambers, theus P(dying) = 1/2. So you go for the respin and reduce P(dying) to 1/3.

  5. @srcav permalink

    And if you don’t know, you respin as P(dying) without would be 2/5 which is more than 1/3

  6. Anonymous permalink

    Following Dave Gale’s point that this more about a discussion than an answer, I have decided to repost my response (non-encrypted this time):You should re-spin. The fact the other player has struck an empty chamber reduces your chances of striking an empty chamber by 1. If you do not re-spin, you will have a 3/5 (0.6) chance of striking an empty chamber and 2/5 (0.4) chance of striking an loaded chamber. With a re-spin, you increase your odds of striking an empty chamber from 3/5 (0.6) to 4/6 (0.666 recurring).Knowing whether or not the bullets are adjacent or not has no impact on your chances when the other player has struck an empty chamber. As you only know the last chamber to be struck was empty, you have no idea whether or not the next chamber contains a bullet.If the other player had struck a chamber containing a bullet and for whatever reason you decided to play on, if the bullets are adjacent, you would have a 1/2 (0.5) chance that the next chamber would either contain a bullet or not (either the next chamber or the previous chamber would have to contain the other bullet if they are adjacent). In this instance, you would be better off re-spining, as this would change your chance of striking an empty chamber from 1/2 (0.5) to 5/6 (0.833 recurring).If you didn’t know if the bullets are adjacent or not, you would have a 4/5 (0.8) chance of striking an empty chamber. Again, you would be better off re-spinning as this would increase your chances of getting an empty chamber to 5/6 (0.833 recurring).

  7. Anonymous permalink

    In response to @srcav, I believe you may have made an error in your calculations. I am ready to be corrected, but I believe I am right in saying the following.You state that: "The reason it is 1/4 is the two chambers immediately after bullets (a bullet and an empty) cant possibly be the next up, as you would have needed a bullet. This leaves you 4 possible chambers you could be on, one of these 4 contains a bullet." However, the fact that your opponent has fired an empty chamber surely reduces the number of possible chambers to 3 (you cannot be on the empty chamber that has just been struck). This reduces the odds to 1/3 of the next chamber containing a bullet. Which is the same odds as re-spinning.So while my originally assessment was incorrect (I originally stated that you would be better off re-spinning), whether you spin or not does not actually change your odds.Effectively, spinning or not does not change your odds. So one could argue we are both right in our suggestions, despite them being the exact opposite, as they both result in the same probability.

  8. Anonymous permalink

    @Dave Gale, before I read @srcav’s response, I didn’t really understand what you meant. Following @srcav, I assume you mean that the blank provides you with information about the probably of what is in the next chamber.

  9. @srcav permalink

    Max, You can’t be on the empty chamber that has just been stuck, no. That leaves 3 possible empty chambers you could be on, and you could be on the first bullet, thus P(dying) is 1/4, as i said originally.

  10. @srcav permalink

    I’m pretty sure i explictly said you can’t be on the first empty chamber ( "The reason it is 1/4 is the two chambers immediately after bullets (a bullet and an empty) cant possibly be the next up" ). And thus there are 4 possibly chambers one of which is a bullet, thus P(dying) = No. of chambers with bullet/Number of chambers possible = 1/4. A respin give 2/6 so 1/3. Thus it is definitely better to respin as a probabiulity of dying of 1/4 is much preferably to a probability of dying of 1/3.The fact you know they are adjacent definitely affects the problem as if know they’re not adjacent don’t know then the P(dyin) in 1/2 so you’d definitley want to respin.I think you have made errors in your calculations, as the respin definitely affects the odds in each case.If you don’t know if they are or not, then you’d respin also, as P (dying) is 2/5 without a respin and 1/3 with.

  11. Anonymous permalink

    @srcav, I said I was prepared to be corrected and I duly consider myself to be corrected by you. :)I hope you took no offence to my (incorrect) attempt to correct you (no offence was intended).Having read your additional comments, I now see my error. I forgot to consider that the empty chamber fired by your opponent could be the empty chamber that directly follows the second bullet and therefore is already excluded by the fact that a bullet has not been fired. I was originally thinking that you were suggesting there were four possible chambers left, but that you had not factored in that one of those four could be the one that was just struck, which would reduce number to three.I hope this clarifies my thinking.Your statement of "you can’t be on the first empty chamber", did raise an interesting point in my mind. I do not wish to sound pedantic for the sake of it, but I would suggest that the use of "first empty chamber" is open to ambiguity.Again, I say this only as a point of discussion, with no offence intended.I assume you mean the first empty chamber following the loaded chambers in a clockwise direction (as I assume that is the direction the chambers would rotate). However, of course you could also be referring to the first empty chamber before the loaded chambers instead, since first empty chamber depends both on where you start from and which direction (clockwise or anti-clockwise) you rotate the chambers. In fact, one could argue any one of the empty chambers is the first.This then lead me to an idea I probably should have used in the beginning, which I have now used to clearly explain to myself as much as anyone else, the different probabilities of each scenario. With hindsight, I should have known the knowledge of whether or not the bullets are adjacent or not would make a difference, otherwise the question would almost certainly have not been asked (of course, there was always the possibility it was a trick question).I have posted my break-down in my subsequent post. Thank you for taking the time to explain your answer to me.

  12. Anonymous permalink

    The following is a break down of all three possible combinations; knowning the bullets are adjcent, knowning they are not adjcent or not knowing whether they are adjcent or not).Adjacent Loaded Chamber Combinations:EEEELLEEELLEEELLEEELLEEELLEEEELEEEEL4/6 or 2/3 Empty2/6 or 1/3 LoadedWe know the combination on our opponent’s turn (if an empty chamber struck) must have been one of:EEEELLEEELLEEELLEEELLEEETherefore, on our turn (without spinning) the possible combinations are:EEELLEEELLEEELLEEELLEEEE3/4 Empty1/4 LoadedTherefore, on our turn (after spinning) the possible combinations are:EEEELLEEELLEEELLEEELLEEELLEEEELEEEEL4/6 or 2/3 Empty2/6 or 1/3 LoadedConclusion: When you know the bullets are adjacent and your opponent strikes an empty chamber, do not spin.Non-Adjacent Loaded Combinations:ELELEEELEELEELEEELEELELEEELEELEEELELLELEEELEELEELEEELE6/9 or 2/3 Empty3/9 or 1/3 LoadedWe know the combination on opponent’s turn (if empty chamber struck) must have been one of:ELELEEELEELEELEEELEELELEEELEELEEELELTherefore, on our turn (without spinning) the possible combinations are:LELEEELEELEELEEELEELELEEELEELEEELELE3/6 or 1/2 Empty3/6 or 1/2 LoadedTherefore, on our turn (after spinning) the possible combinations are:ELELEEELEELEELEEELEELELEEELEELEEELELLELEEELEELEELEEELE6/9 or 2/3 Empty3/9 or 1/3 LoadedConclusion: When you know the bullets are not adjacent and your opponent strikes an empty chamber, spin.Adjecny Unknown Combinations:EEEELLEEELLEEELLEEELLEEEELELEEELEELEELEEELEELELEEELEELEEELELLELEEELEELEELEEELELLEEEELEEEEL10/15 or 2/3 Empty 5/15 or 1/3 LoadedWe know the combination on Opponent’s Turn (if empty chamber struck) must have been one of:EEEELLEEELLEEELLEEELLEEEELELEEELEELEELEEELEELELEEELEELEEELELTherefore, on our Turn (without spinning) the possible combinations are:EEELLEEELLEEELLEEEELELEEELEELEEELELELLEEEELELEEELEELEELEEELE6/10 or 3/5 Empty4/10 or 2/5 LoadedTherefore, on our turn (after spinning) the possible combinations are:EEEELLEEELLEEELLEEELLEEEELELEEELEELEELEEELEELELEEELEELEEELELLELEEELEELEELEEELELLEEEELEEEEL10/15 or 2/3 Empty 5/15 or 1/3 LoadedConclusion: When you don’t know whether the bullets are not adjacent and your opponent strikes an empty chamber, spin.

  13. @srcav permalink

    Hi Max, No offence was taken, sorry if I implied it in my m,essage. You are corred that the "first empty chamber" is slightly ambiguos, i did mean the first one after the bullets!!

  14. Anonymous permalink

    Hello @srcav,I’m glad to know I didn’t offend you. I didn’t think I had, but I wasn’t sure (it’s obviously harder to judge how something is being said when it’s only in written form), so it’s nice to know. Thank you.

  15. @srcav permalink

    Indeed, the problem with text as a medium for discussion is that sometimes tone is hard to read, without the voice and facial expression text is just cold words and can often be taking in a variety of ways… I guess this is the sort of thing English teachers deal with!

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