# Odds problem - roll a pair of dice **URL:** \ **Category:** Factual Questions\ **Created:** [October 6, 2016, 1:41pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763 "2016-10-06T13:41:48Z")\ **Posts on this page:** 20\ **Page:** 20
**Author:** ![sich\_hinaufwinden](https://avatars.discourse-cdn.com/v4/letter/s/c5a1d2/32.png) [@sich\_hinaufwinden](https://boards.straightdope.com/u/sich_hinaufwinden)\ **Post date:** [October 10, 2016, 7:44pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/381 "2016-10-10T19:44:49Z")
> [@LSLGuy](#): > > @ **sich\_hinaufwinden** : > I’m in the sorta 1/6 camp. I’ve posited a particular interpretation that gives 1/6. See [http://boards.straightdope.com/sdmb/showpost.php?p=19688130&postcount=346](http://boards.straightdope.com/sdmb/showpost.php?p=19688130&postcount=346) on the previous page for one example… I remember way early in the thread you mentioned that you favored a 1/6 interpretation. At the time your comment made me think to myself that I favored the 1/11 interpretation because I think it is the more interesting scenario and analysis. But I’ve come around to thinking that at least one type of 1/6 scenario requires the least amount of assumptions. > [@LSLGuy](#): > > …But I’ve always caveated that my answer depends on my assumptions. Which are merely reasonable assumptions not prohibited by the strict text. > > I mostly take this 1/6 tack to poke at the “It’s gotta be 1/11; there’s no other possibility period, foot stamp, hold my breath.” camp… Which is why I would have thought you’d consider yourself in the “not enough info to give a single answer” camp. Doesn’t really matter much I guess - the comment I responded to made it sound like there were lots of people in a “1/6 and 1/6 only” camp and that didn’t fit my perception so I questioned it. ---
**Author:** ![x-ray\_vision](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/x-ray_vision/32/351_2.png) [@x-ray\_vision](https://boards.straightdope.com/u/x-ray_vision)\ **Post date:** [October 10, 2016, 7:56pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/382 "2016-10-10T19:56:42Z")
> [@Chronos](#): > > **x-ray-vision** , under what circumstances would we have that information? The only circumstance that would give an answer of 1/3 would be something like a census worker going through the list of two-children families, throwing out all of the families where they’re both girls, and then picking one of the remaining families at random… _but nobody ever actually does that_. The post of yours I was responding to started with you saying: \*I would argue that the only reasonable interpretation of the boy-girl problem is the one which yields 1/2. \* It’s “The boy-girl problem.” It’s a specific question asked specific way, and it doesn’t have only one reasonable interpretation you claim it does, especially for the reason you claim that is so. Understanding how to solve these problems have real-world applications; it’s irrelevant that the specific situations within the problems themselves will probably never arise within one’s lifetime. > [@](#): > > Realistically, the way that we know that one of Mr. Smith’s children is a boy is that we’ve met one of them, and in that case, the answer is 1/2 for the other one. Okay, **in that case**. But the phrasing of the boy-girl problem has a different case than a completely different case you think it should be. As a reminder, this thread started with “A math teacher friend of mine shared this problem…” And everything Thudlow Boink said. ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [October 10, 2016, 9:02pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/383 "2016-10-10T21:02:22Z")
> [@LSLGuy](#): > > The whole puzzle is (arguably) equivalent to rolling one die until you’ve rolled a six, however many times that takes. Then rolling the other die and asking “what are the odds the second one comes up six?” To which the answer is 1/6. > > Rolling them simultaneously doesn’t change anything. What changes things is what the observer says and under exactly what circumstances he/she says it. No! Rolling both at the same time and rolling them sequentially are _very_ different. The OP is more like roll both dice until at least one of them is a 6, what are the chances that the other is also a 6? It takes fewer rolls on average to get 1+ 6 out of two dice than a 6 out of 1 dice. ---
**Author:** ![The\_Great\_Unwashed](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_great_unwashed/32/4791_2.png) [@The\_Great\_Unwashed](https://boards.straightdope.com/u/The_Great_Unwashed)\ **Post date:** [October 10, 2016, 9:26pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/384 "2016-10-10T21:26:14Z")
> [@ftg](#): > > No! Rolling both at the same time and rolling them sequentially are _very_ different. How so? Before you furnish your answer, you may wish to reflect on the fact that it’s physically impossible to throw two dice at \*exactly \*the same time. ---
**Author:** ![Haldurson](https://avatars.discourse-cdn.com/v4/letter/h/51bf81/32.png) [@Haldurson](https://boards.straightdope.com/u/Haldurson)\ **Post date:** [October 11, 2016, 12:54am UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/385 "2016-10-11T00:54:43Z")
> [@Musicat](#): > > There’s your problem. No die has a ten on any face. It says “You are ten”, it doesn’t say that the die shows a ten. But its 1 out of 6. It doesn’t matter what was on the other die, it’s 1 out of 6. If you are rolling a six-sided die, and there’s a six on exactly one face of that die, then it is 1 out of 6. That is, unless someone played a dirty trick on you and replaced the 6 with a different number. You may as well ask what the chances are of rolling a 6 on a die if you are in the 6th row of the number 6 bus going down 6th avenue, you are six minutes late for work on the 6th floor of room 666, and its 6/6/66. They are all independent of one another. ---
**Author:** ![octopus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/octopus/32/3716_2.png) [@octopus](https://boards.straightdope.com/u/octopus)\ **Post date:** [October 11, 2016, 1:23am UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/386 "2016-10-11T01:23:48Z")
> [@Haldurson](#): > > It says “You are ten”, it doesn’t say that the die shows a ten. > > But its 1 out of 6. It doesn’t matter what was on the other die, it’s 1 out of 6. If you are rolling a six-sided die, and there’s a six on exactly one face of that die, then it is 1 out of 6. That is, unless someone played a dirty trick on you and replaced the 6 with a different number. > > You may as well ask what the chances are of rolling a 6 on a die if you are in the 6th row of the number 6 bus going down 6th avenue, you are six minutes late for work on the 6th floor of room 666, and its 6/6/66. They are all independent of one another. Roll some dice as described in the OP and come back in an hour . ---
**Author:** ![sich\_hinaufwinden](https://avatars.discourse-cdn.com/v4/letter/s/c5a1d2/32.png) [@sich\_hinaufwinden](https://boards.straightdope.com/u/sich_hinaufwinden)\ **Post date:** [October 11, 2016, 1:38am UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/387 "2016-10-11T01:38:09Z")
> [@octopus](#): > > Roll some dice as described in the OP and come back in an hour . The OP doesn’t sufficiently describe how to go about playing this game for an hour. We’ve only been told about a single roll that can be interpreted in more than one way. I’m not defending the logic in the post you quote - just pointing out that he can’t do what you suggest. ---
**Author:** ![misterdls](https://avatars.discourse-cdn.com/v4/letter/m/f4b2a3/32.png) [@misterdls](https://boards.straightdope.com/u/misterdls)\ **Post date:** [October 11, 2016, 1:44am UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/388 "2016-10-11T01:44:47Z")
This question is presented in a confusing manner, but is very easy, if one properly understands sample spaces. When 2 six sided dice, 1 Green and 1 Red, are rolled, and you want to consider only those outcomes that include either die showing a 6, there are only 11 possible results: G6 R1 G6 R2 G6 R3 G6 R4 G6 R5 G6 R6 G1 R6 G2 R6 G3 R6 G4 R6 G5 R6 This is the only proper sample space. Only 1 of the 11 outcomes allows you to “pick a 6” AND see a 6 on the other die. Therefore the answer to the original question is 1/11. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [October 11, 2016, 2:02am UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/389 "2016-10-11T02:02:47Z")
That _might_ be the only proper sample space, depending on the details of the problem which we aren’t given. ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [October 11, 2016, 2:29am UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/390 "2016-10-11T02:29:51Z")
> [@sich\_hinaufwinden](#): > > … > Which is why I would have thought you’d consider yourself in the “not enough info to give a single answer” camp. > > Doesn’t really matter much I guess - the comment I responded to made it sound like there were lots of people in a “1/6 and 1/6 only” camp and that didn’t fit my perception so I questioned it. My deep attitude is the “not enough info to give a single answer” camp. My negotiating tactic in the thread has been to consistently give examples in the 1/6 camp while explaining that that’s a choice, not a foregone conclusion. And an arguable choice at that. I picked the minority approach hoping to coax the majority folks off their “It’s gotta be 1/11 there’s no other possibility!” bandwagon. Once they can admit there are two possibilities, then it’s time to stick the wedge in the crack and pound away until we get awareness of the full panoply of choices. Of admittedly varying plausibility. One of the resident professional math experts (\*\*Chronos \*\* I believe) has laid this full line of reasoning out several times. And been mostly on ignore by everyone else. To my ongoing amazement. ---
**Author:** ![octopus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/octopus/32/3716_2.png) [@octopus](https://boards.straightdope.com/u/octopus)\ **Post date:** [October 11, 2016, 4:01am UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/391 "2016-10-11T04:01:17Z")
> [@sich\_hinaufwinden](#): > > The OP doesn’t sufficiently describe how to go about playing this game for an hour. > > We’ve only been told about a single roll that can be interpreted in more than one way. > > I’m not defending the logic in the post you quote - just pointing out that he can’t do what you suggest. Why can’t he? It would be empirical confirmation of the analytical answer of 1/11. I don’t know why folks are making the question out to be some form of semantic gotcha. What’s so puzzling about a pair of fair dice were rolled and one person was shown a 6? ---
**Author:** ![octopus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/octopus/32/3716_2.png) [@octopus](https://boards.straightdope.com/u/octopus)\ **Post date:** [October 11, 2016, 4:02am UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/392 "2016-10-11T04:02:44Z")
> [@Chronos](#): > > That _might_ be the only proper sample space, depending on the details of the problem which we aren’t given. What details are sufficient to satisfy everyone who may claim the question is ambiguous? Can you write it in such a way that no one could claim ambiguity? ---
**Author:** ![sich\_hinaufwinden](https://avatars.discourse-cdn.com/v4/letter/s/c5a1d2/32.png) [@sich\_hinaufwinden](https://boards.straightdope.com/u/sich_hinaufwinden)\ **Post date:** [October 11, 2016, 4:22am UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/393 "2016-10-11T04:22:05Z")
> [@octopus](#): > > Why can’t he?.. You told him to roll dice for an hour. Presumably that is more than one roll and some of those rolls will not have at least one six. What happens on those rolls? And how do you know that? ---
**Author:** ![BigT](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bigt/32/12044_2.png) [@BigT](https://boards.straightdope.com/u/BigT)\ **Post date:** [October 11, 2016, 4:22am UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/394 "2016-10-11T04:22:57Z")
> [@Chronos](#): > > I’ll join in with the “problem is ambiguous” crowd, here. There are multiple possible experiments that are all consistent with the problem statement, and which have different answers. Here are two examples: > > 1: I roll a pair of dice. I then choose one die at random, look at the number on the die, and announce “At least one die is a \_\_\_”, where the blank is the number I looked at. On this particular iteration of the experiment, the die I looked at was a 6, and I announce as such. What is the probability that the other die is a 6? > > Answer: The probability that the other die is a 6 is 1 in 6… This is adding information not stated in the problem. At no point does it say that they picked one die at random and read the number. The number 6 is given in the problem without any means for it to be different, and thus is fixed. Any simulation must keep the 6, or else it is modeling a different problem. And once you do require there to be a 6, the 1/11 interpretation is the only one that works. I do not understand why such an explanation is still being promoted, when one of the basic rules of word problems is that all required information is given. ---
**Author:** ![octopus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/octopus/32/3716_2.png) [@octopus](https://boards.straightdope.com/u/octopus)\ **Post date:** [October 11, 2016, 4:40am UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/395 "2016-10-11T04:40:54Z")
> [@sich\_hinaufwinden](#): > > You told him to roll dice for an hour. > > Presumably that is more than one roll and some of those rolls will not have at least one six. > > What happens on those rolls? > > And how do you know that? You can’t be shown a 6 and asked the probability of the other die being a 6 if the roll doesn’t include a 6. ---
**Author:** ![sich\_hinaufwinden](https://avatars.discourse-cdn.com/v4/letter/s/c5a1d2/32.png) [@sich\_hinaufwinden](https://boards.straightdope.com/u/sich_hinaufwinden)\ **Post date:** [October 11, 2016, 4:41am UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/396 "2016-10-11T04:41:25Z")
> [@octopus](#): > > What details are sufficient to satisfy everyone who may claim the question is ambiguous? Can you write it in such a way that no one could claim ambiguity? A person rolls a pair of dice until it is possible for him to reveal a 6 at which point he reveals it. Then and only then he asks what the probability is that the other die is a 6. I think that would be good enough for me, if they also make 1/11 one of the answer choices. I don’t think things like specifying 6-sided dice and that they are fair are what the discussion is about at this point. This isn’t about semantic gotchas or the like - it’s about a legitimate lack of detail necessary to narrow the solution down to a single answer. ---
**Author:** ![octopus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/octopus/32/3716_2.png) [@octopus](https://boards.straightdope.com/u/octopus)\ **Post date:** [October 11, 2016, 4:43am UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/397 "2016-10-11T04:43:57Z")
> [@sich\_hinaufwinden](#): > > A person rolls a pair of dice until it is possible for him to reveal a 6 at which point he reveals it. > > Then and only then he asks what the probability is that the other die is a 6. > > I think that would be good enough for me, if they also make 1/11 one of the answer choices. > > I don’t think things like specifying 6-sided dice and that they are fair are what the discussion is about at this point. > > This isn’t about semantic gotchas or the like - it’s about a legitimate lack of detail necessary to narrow the solution down to a single answer. How can you show a 6 if one isn’t rolled? Again, making the assumption that people are being honest, etc. And supposedly the OP’s question is modeled after a different question in which a 4 is shown. In which case there is a correct answer, 2/11, as a choice. ---
**Author:** ![sich\_hinaufwinden](https://avatars.discourse-cdn.com/v4/letter/s/c5a1d2/32.png) [@sich\_hinaufwinden](https://boards.straightdope.com/u/sich_hinaufwinden)\ **Post date:** [October 11, 2016, 4:59am UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/398 "2016-10-11T04:59:24Z")
> [@octopus](#): > > How can you show a 6 if one isn’t rolled? Again, making the assumption that people are being honest, etc. > > And supposedly the OP’s question is modeled after a different question in which a 4 is shown. In which case there is a correct answer, 2/11, as a choice. You asked what details were necessary to satisfy everyone about ambiguity so I gave my input. I want it explicitly stated that he is throwing out 25/36 of the rolls by only asking about a pre-determined number. Why wouldn’t you state that explicitly if it is the case? It doesn’t hurt anything to state it that way does it? Why frame the question in a way that can leave people in doubt about the rules of the game? ---
**Author:** ![Cugel](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cugel/32/1199_2.png) [@Cugel](https://boards.straightdope.com/u/Cugel)\ **Post date:** [October 11, 2016, 5:28am UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/399 "2016-10-11T05:28:59Z")
> [@sich\_hinaufwinden](#): > > it’s about a legitimate lack of detail necessary to narrow the solution down to a single answer. Nah. Pretty clear the spirit of the problem is to get people to guess the “obvious” 1/6, instead of the actual 1/11. ---
**Author:** ![octopus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/octopus/32/3716_2.png) [@octopus](https://boards.straightdope.com/u/octopus)\ **Post date:** [October 11, 2016, 6:25am UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/400 "2016-10-11T06:25:45Z")
> [@sich\_hinaufwinden](#): > > You asked what details were necessary to satisfy everyone about ambiguity so I gave my input. > > I want it explicitly stated that he is throwing out 25/36 of the rolls by only asking about a pre-determined number. > > Why wouldn’t you state that explicitly if it is the case? It doesn’t hurt anything to state it that way does it? Why frame the question in a way that can leave people in doubt about the rules of the game? Because showing a 6 or a 2 or a 4 is enough to know that that particular number was in fact shown. It’s a given. If I say I have 3 black socks, 1 gray sock, and 1 paisley sock in my sock drawer and I draw one black one completely randomly what does anything else matter if the question then asks what’s the probability that the next sock randomly drawn is gray? At some point normal assumptions should be sufficient. I’m just trying to imagine getting through an engineering curriculum with this standard of precision required. [Previous page](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763.md?page=19) [Next page](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763.md?page=21)