# 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:** 22
**Author:** ![Cliffy\_Biro](https://avatars.discourse-cdn.com/v4/letter/c/57b2e6/32.png) [@Cliffy\_Biro](https://boards.straightdope.com/u/Cliffy_Biro)\ **Post date:** [October 11, 2016, 5:43pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/421 "2016-10-11T17:43:52Z")
> [@Thudlow\_Boink](#): > > Ah. What was the original problem? (My apologies if it has already been mentioned in this thread.) The [next](https://brilliant.org/practice/conditional-probability-level-4-5-challenges/?subtopic=probability-2&chapter=conditional-probability) puzzle at that site is analagous to the OP’s and the boy/girl paradox, containing as it does the dread phrase “I announce that at least one…” : \*One green ball, one blue ball, and two red balls are placed in a bowl. I draw two balls simultaneously from the bowl and announce that at least one of them is red. What is the chance that the other ball I have drawn out is also red?\* Under the two most common assumptions used in this thread, the answer is either 1/5 or 1/3. Both answers are among the choices. I’m not going to sign up to that site to see which answer they want! ---
**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 11, 2016, 5:47pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/422 "2016-10-11T17:47:33Z")
> [@Cliffy\_Biro](#): > > The [next](https://brilliant.org/practice/conditional-probability-level-4-5-challenges/?subtopic=probability-2&chapter=conditional-probability) puzzle at that site is analagous to the OP’s and the boy/girl paradox, containing as it does the dread phrase “I announce that at least one…” : > > \*One green ball, one blue ball, and two red balls are placed in a bowl. I draw two balls simultaneously from the bowl and announce that at least one of them is red. > > What is the chance that the other ball I have drawn out is also red?\* > > Under the two most common assumptions used in this thread, the answer is either 1/5 or 1/3. Both answers are among the choices. I’m not going to sign up to that site to see which answer they want! That looks like a sock puzzle I once invented. Plagiarists! ---
**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, 6:04pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/423 "2016-10-11T18:04:30Z")
> [@The\_Great\_Unwashed](#): > > That looks like a sock puzzle I once invented. Plagiarists! You used a sock account to post a puzzle? Reported! ---
**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, 6:08pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/424 "2016-10-11T18:08:43Z")
> [@Cliffy\_Biro](#): > > The [next](https://brilliant.org/practice/conditional-probability-level-4-5-challenges/?subtopic=probability-2&chapter=conditional-probability) puzzle at that site is analagous to the OP’s and the boy/girl paradox, containing as it does the dread phrase “I announce that at least one…” : > > \*One green ball, one blue ball, and two red balls are placed in a bowl. I draw two balls simultaneously from the bowl and announce that at least one of them is red. > > What is the chance that the other ball I have drawn out is also red?\* > > Under the two most common assumptions used in this thread, the answer is either 1/5 or 1/3. Both answers are among the choices. I’m not going to sign up to that site to see which answer they want! Time to get some balls and a bowl! ---
**Author:** ![newme](https://avatars.discourse-cdn.com/v4/letter/n/e0b2c6/32.png) [@newme](https://boards.straightdope.com/u/newme)\ **Post date:** [October 11, 2016, 6:09pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/425 "2016-10-11T18:09:08Z")
You are walking through the commons at the university. You see a student wearing a t-shirt with “1/11” emblazoned on the front, standing beside a table and holding a tabulation chart. Other students seem to be giving him a wide berth. Curious, you approach him. “I am doing a study in probability, where if one die is known, what is the likely result of the other die” he says. “Please roll this pair of fair, six-sided dice.” You roll, and see the result is 3 and 6. You casually remark," There is at least one 3. " “You’re wrong. You should have said there is at least one six. I was only looking for sixes,” he says. “I picked a number at random. You never told me you were looking for sixes,” you say. “You should have assumed it. To assume otherwise is ABSURD!” (You back away slowly.) ---
**Author:** ![Cliffy\_Biro](https://avatars.discourse-cdn.com/v4/letter/c/57b2e6/32.png) [@Cliffy\_Biro](https://boards.straightdope.com/u/Cliffy_Biro)\ **Post date:** [October 11, 2016, 6:13pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/426 "2016-10-11T18:13:10Z")
> [@octopus](#): > > Time to get some balls and a bowl! Time to get a bowl of popcorn and a deckchair, more like. This could set the whole thing of again. ---
**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, 6:22pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/427 "2016-10-11T18:22:39Z")
This is a trivial math question and yet people are having so many problems with it – I don’t get it. I’ll quote the original post so that there is no confusion: “Two fair dice are rolled together, and you are told that ‘at least one of the dice is a 6.’ A 6 is removed, and you are ten shown the other die. What is the probability of the other die showing a six?” Translation: You roll two dice, and at least one of them is a 6. Given that the first die is a six, what is the probability that the second die is a 6. There’s a red herring here, and that is the first part of the question. If you roll one die, the chances of rolling a 6 is one out of 6. Everything else is irrelevant. The first part of the question is rendered irrelevant by the revelation that the first die is a 6, because that satisfies the requirement that at least one die is a 6. So the question comes down to what is the probability of rolling a 6 on a single die. and it is 1 out of 6. It really should not be this confusing to anyone. ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [October 11, 2016, 6:24pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/428 "2016-10-11T18:24:43Z")
> [@Some\_Call\_Me.Tim](#): > > This probably won’t help anyone but bridge players already familiar with it, but this is similar to the [principle of restricted choice](https://en.wikipedia.org/wiki/Principle_of_restricted_choice). This thread has gotten so long we’re repeating ourselves! > [@septimus](#): > > As I said upthread, this puzzle is in the Monty Hall category. Several weeks ago I posted another such > To avoid hijacking this thread I’ll Spoiler the problem statement. > > [SPOILER]The puzzle arises in the game of contract bridge, but non-players should be able to understand the setup. You and your partner (the dummy) have nine hearts missing only the Queen, Jack, Trey and Deuce. You cash the Ace; LHO plays the Trey, RHO the Jack. You lead toward dummy’s King, LHO following with the Deuce. _Should you play for the Queen to be with LHO or RHO?_ (Your opponents are indifferent in their plays of Trey vs Deuce, or Queen vs. Jack. However, as the play has proceeded they would not play a picture card when a small card still available.) > > There are only two possibilities that remain for the original holdings: > > LHO: 3 2 RHO: Q J > LHO: Q 3 2 RHO: J > > The cards were shuffled randomly to begin with and, with nothing else to go on, the _a priori_ chances for these two cases are about the same. (1st case is slightly more likely than 2nd.) > _What is the percentage play now?_ > > [/SPOILER] ---
**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, 6:28pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/429 "2016-10-11T18:28:11Z")
> [@Haldurson](#): > > This is a trivial math question and yet people are having so many problems with it – I don’t get it. I’ll quote the original post so that there is no confusion: > “Two fair dice are rolled together, and you are told that ‘at least one of the dice is a 6.’ A 6 is removed, and you are ten shown the other die. What is the probability of the other die showing a six?” > > Translation: You roll two dice, and at least one of them is a 6. Given that the first die is a six, what is the probability that the second die is a 6. > > There’s a red herring here, and that is the first part of the question. If you roll one die, the chances of rolling a 6 is one out of 6. Everything else is irrelevant. The first part of the question is rendered irrelevant by the revelation that the first die is a 6, because that satisfies the requirement that at least one die is a 6. So the question comes down to what is the probability of rolling a 6 on a single die. and it is 1 out of 6. It really should not be this confusing to anyone. Doesn’t quite work like that. Get some dice and try 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 11, 2016, 6:50pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/430 "2016-10-11T18:50:49Z")
> [@octopus](#): > > Doesn’t quite work like that. Get some dice and try it. It does work like that given his translation. It’s 1/6. ---
**Author:** ![Nelsterm](https://avatars.discourse-cdn.com/v4/letter/n/b9bd4f/32.png) [@Nelsterm](https://boards.straightdope.com/u/Nelsterm)\ **Post date:** [October 11, 2016, 7:35pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/431 "2016-10-11T19:35:23Z")
> [@octopus](#): > > Doesn’t quite work like that. Get some dice and try it. It does though. If you have two dice called die A and die B and you know only that at least one of them is a six the chances of both being six from what you know at that time is 1 in 11 because we don’t know whether die A or B is definitely a six. Once either die A or B is removed as definitely being a six we have more information. We have specified one particular instance of the dice as being a six and so cut down the remaining possibilities. Once we know that we can deduce that the chance of the other instance of a die being a six is 1 in 6. These are two different questions. ---
**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, 7:57pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/432 "2016-10-11T19:57:23Z")
> [@Cliffy\_Biro](#): > > Time to get a bowl of popcorn and a deckchair, more like. This could set the whole thing of again. It’s especially delicious that the very next post after yours is **Haldurson** walking in for a fresh whack at it as if he was poster #2. With **octopus** and **x-ray vision** in hot pursuit on their respective hobby horses. We’re now at post #430. Do we think **Haldurson** _et al_ can power us up for another orbit up to #860? Of course we do. Onwards and upwards! Someone is wrong on the internet and we shall not rest until Error is Vanquished and Righteousness is Restored! I promise to do my bit in turn. 🙂 😃 ---
**Author:** ![Colibri](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/colibri/32/1841_2.png) [@Colibri](https://boards.straightdope.com/u/Colibri)\ **Post date:** [October 11, 2016, 8:15pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/433 "2016-10-11T20:15:39Z")
> [@LSLGuy](#): > > It’s especially delicious that the very next post after yours is **Haldurson** walking in for a fresh whack at it as if he was poster #2. With **octopus** and **x-ray vision** in hot pursuit on their respective hobby horses. > > We’re now at post #430. Do we think **Haldurson** _et al_ can power us up for another orbit up to #860? Of course we do. Onwards and upwards! Someone is wrong on the internet and we shall not rest until Error is Vanquished and Righteousness is Restored! > > I promise to do my bit in turn. 🙂 😃 **Moderator Note** Let’s do so with less snark. No warning issued, but make your points without getting personal. Colibri General Questions Moderator ---
**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, 8:40pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/434 "2016-10-11T20:40:48Z")
My apologies to one and all. I’m at least as guilty of riding my hobby horse as anyone else here. I regret that my intent was not as clear as it should have been. ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [October 11, 2016, 9:51pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/435 "2016-10-11T21:51:39Z")
How about this phrasing of the puzzle? Alice rolls a pair of dice, so that Bob can see the outcome but Alice can’t. She asks Bob, “Is there at least one six?” Bob replies “yes.” What is the probability that the other is a six? I believe this is unambiguous, and the answer is 1/11. This phrasing is also parallel to the two-child puzzle, in that it’s the census worker who doesn’t know and asks the “either one” question. > [@BigT](#): > > 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. It doesn’t state that, but it is a semi-reasonable assumption for what happened based on the wording. Sure, the more common assumption would be that he saw both and just playfully decided to say the enigmatic “at least one is a six” phrase, but I can see how someone may not go with that assumption. In that way, the wording of the puzzle in the OP is similar to the typical wording of the Monty Hall puzzle in that we need more information. ETA: In my restatement of the puzzle above, I guess it would also be possible for Alice to take the simple “yes” as meaning that there is only one six, because if there were two, then Bob would have said “as a matter of fact, they both are.” That’s what would happen in real life, and in that case the probability is zero. So you’d have to modify it so that Alice tells Bob she wants only a yes or no answer: “Yes or no: is at least one of them a six?” ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [October 11, 2016, 10:10pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/436 "2016-10-11T22:10:22Z")
> [@LSLGuy](#): > > My apologies to one and all. > > I’m at least as guilty of riding my hobby horse as anyone else here. > > I regret that my intent was not as clear as it should have been. I don’t have the time to review your posts. If you’re troubled over your alignment, simply compare it with the exemplary posts of **septimus** which cover the issues thoroughly, correctly and continently! 😃 :smack: > [@CurtC](#): > > How about this phrasing of the puzzle? > > Alice rolls a pair of dice, so that Bob can see the outcome but Alice can’t. She asks Bob, “Is there at least one six?” Bob replies “yes.” What is the probability that the other is a six? > > I believe this is unambiguous, and the answer is 1/11. This phrasing is also parallel to the two-child puzzle, in that it’s the census worker who doesn’t know and asks the “either one” question. > > It doesn’t state that, but it is a semi-reasonable assumption for what happened based on the wording. Sure, the more common assumption would be that he saw both and just playfully decided to say the enigmatic “at least one is a six” phrase, but I can see how someone may not go with that assumption. In that way, the wording of the puzzle in the OP is similar to the typical wording of the Monty Hall puzzle in that we need more information. > > ETA: In my restatement of the puzzle above, I guess it would also be possible for Alice to take the simple “yes” as meaning that there is only one six, because if there were two, then Bob would have said “as a matter of fact, they both are.” That’s what would happen in real life, and in that case the probability is zero. So you’d have to modify it so that Alice tells Bob she wants only a yes or no answer: “Yes or no: is at least one of them a six?” And **CurtC** immediately bests me with a superb post, carefully outlining the correct answer and its ambiguity. _Had this been #2 the Mod could have closed the thread then and there!_ As is, I suppose the thread may continue to grow, perhaps becoming like the 1=.9999999999 thread. Should it be moved to IMHO? How about ATMB? :rolleyes: ---
**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, 10:50pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/437 "2016-10-11T22:50:47Z")
Yup, **CurtC** ’s statement of the problem is unambiguous, and has the answer he states. ---
**Author:** ![Cliffy\_Biro](https://avatars.discourse-cdn.com/v4/letter/c/57b2e6/32.png) [@Cliffy\_Biro](https://boards.straightdope.com/u/Cliffy_Biro)\ **Post date:** [October 11, 2016, 10:58pm UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/438 "2016-10-11T22:58:26Z")
> [@CurtC](#): > > How about this phrasing of the puzzle? > > Alice rolls a pair of dice, so that Bob can see the outcome but Alice can’t. She asks Bob, “Is there at least one six?” Bob replies “yes.” What is the probability that the other is a six? > > I believe this is unambiguous, and the answer is 1/11. Yes, as long as we can assume that Bob will always make some reply, and as you say that it will be either “yes” or “no”, but I think that particular assumption is reasonable. The important thing is that he who sees the dice doesn’t get to decide what statement is made, and your version addresses that. It guarantees that Alice will always know whether or not there is at least one six. Not sure why we have Alice rolling the dice when Bob’s the one who can see them, but that’s just a practical matter. I’m just trying to picture Alice throwing the dice with her eyes shut or something? As far as I can see, it could be Bob rolling the dice. ---
**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 12, 2016, 12:20am UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/439 "2016-10-12T00:20:23Z")
Nicely done **CurtC**. It avoids the two problems in the OP and is a straightforward description of events that happened where the important elements can truly be taken to be givens. My opinion is that you don’t need the ETA - it’s your story to tell, and in it you had Bob simply say “Yes”. And I think we can all agree on what Alice meant “by at least one 6” and that she would correctly interpret the simple “Yes” answer. I wonder if any of the 1/11 only crowd will acknowledge how your phrasing elegantly sidesteps the problems with the OP. ---
**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 12, 2016, 1:10am UTC](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763/440 "2016-10-12T01:10:59Z")
> [@sich\_hinaufwinden](#): > > I wonder if any of the 1/11 only crowd will acknowledge how your phrasing elegantly sidesteps the problems with the OP. We’d already figured out that this is what was meant from the first reading. [Previous page](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763.md?page=21) [Next page](https://boards.straightdope.com/t/odds-problem-roll-a-pair-of-dice/767763.md?page=23)