The 1/11 argument depends on all eleven possibilities being equally likely, which cannot be deduced from the problem statement.
In the 1/6 version, it is assumed that the six is twice as likely to come from a roll of [6 6] (because there are two sixes in it rather than just one), giving a result of 2/12, i.e. 1/6. Again, this assumption is not justified by the problem statement.
What is in dispute is the meaning of the given information. If the rules are understood to be : if at least one six is observed then that is announced, then the answer is indeed 1/11.
If on the other hand,* if what is announced is the value showing on an arbitrarily chosen die (eg: the first to stop rolling, or, the red one) which in this case it just happened to be a six*, then the answer is 1/6.
I believe the intention of the question setter was the first interpretation, but whatever, the question is ambiguous.
Does this help illustrate the 1/6 (2/12) argument? (Not that I agree with it.) The bold lines are the equally likely possibilities that are consistent with the event described in the problem statement. There are twelve of them, of which two are [6 6].
Dice Die you learn about
==== ===================
1 6 first
**1 6 second**
2 6 first
**2 6 second**
3 6 first
**3 6 second**
4 6 first
**4 6 second**
5 6 first
**5 6 second**
**6 1 first**
6 1 second
**6 2 first**
6 2 second
**6 3 first**
6 3 second
**6 4 first**
6 4 second
**6 5 first**
6 5 second
**6 6 first**
**6 6 second**
One thing that is not helping is people are referring to post #x without a link.
In threads getting this long it’s getting hard to track down a post by link #.
Simply go to that post, hover your mouse over the #number link at the top of that post, right click copy the link. Then in your post, over the text you want to turn into a link, select it, click on the globe-chain icon, paste in the link and voilà.
(You might note that all the #number links in a thread are the same except for the final postcount number. You could copy any post link in this thread and modify the postcount number for your link. But you might get the number wrong. Which seems to be the case with several people saying “See post #x” and I go to post #x and it doesn’t have anything that’s been asserted. Please actually go to the post, verify it is relevant to your claim and copy the # link.)
Anyway: Like I said. I keep checking this unlinked references and never find anything close to what the poster is asserting in that post.
With all the physicists on the Board, I’m surprised that nobody’s blamed this on Bose Einstein statistics, based on dice being bosons. :eek:
The boy-girl problem is mathematically analogous, but because of the different origin of the problem, there’s a difference in what assumptions are reasonable. I would argue that the only reasonable interpretation of the boy-girl problem is the one which yields 1/2. Why? Because in almost every real-world case where we would learn that at least one of someone’s children is a boy, we learn that by gaining information about one specific child. The only interpretations which allow for gaining that information without it referring to one specific child are contrived betting games, and while people often play contrived betting games with dice or coins, they very seldom do so with children.
I’ll see your two tweens and raise you three toddlers! 
Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?
The only reasonable interpretation of what’s clearly a math problem, brain teaser, etc. results in an answer of 1/2 because children were used, but coins would make another answer reasonable? Interesting take.
A key question in all these things is “Are we evaluating a real-world situation, or are we playing a math game probably involving a camouflaged twist or gotcha?” The definition of “reasonable assumptions” will differ in the two scenarios.
The math remains the same. Once you (any you) are done deciding, based on the situation, what to assume. It helps also to tell folks whether you’re assuming a math game or the real world.
Based on my interpretation, “at least one of the dice is a 6” is a given. It’s part of the setup of the problem. To ask “What if neither of the dice had been a 6” is to fight the hypothetical.
Which I think goes to the following issue, of whether you’re thinking pure-mathematically or real-worldily:
LSLGuy makes an excellent point. Are we thinking of this as something happening in the Real World, and we are allowed to bring to bear what we know about the way the real world works? Or are we treating this as a hypothetical mathematical problem in which we are to take the given parameters at face value and not make assumptions that go beyond them? My own instinct is toward the latter.
I am reminded of the old joke:
A mathematician, a physicist, and an engineer are on a train going through Scotland. The engineer sees a black sheep, and says, “Aha! The sheep in Scotland are black!” The physicist shakes his head and says, “Ha! You’re wrong! One sheep in Scotland is black!” The mathematician shakes his head sadly and says, “You’re both wrong. One sheep in Scotland is black on one side…”
But how did it come to be a given? I think that is important to make a meaningful calculation of the probability.
If all I know is that based on a single roll of the dice, one is shown and I’m asked what the probability is the other one is the same number, it could look to me like the person is going to always show me a die and ask me about doubles after every roll.
I haven’t been told any rules, haven’t heard about any target number, haven’t seen any evidence that rolls would be thrown out if they didn’t include a particular number.
All I know is one roll, one reveal, one question.
So it doesn’t matter what die is revealed, the question is whether or not doubles were rolled, and that happens 1/6 of the time.
I think that’s an equally logical interpretation of the events described in the OP.
I mean if he reveals a 3, he’s not going to ask what the chances are he rolled double 5s. So since he has decided to reveal the 3, he has to ask about double 3s and says “I’ve rolled at least one 3 and here I am showing it to you - what are the chances the other one is a 3 as well?”
If I know or suspect that he is going to ask about some kind of doubles after every roll, the chances are 1/6 that he rolled doubles.
And I don’t think one roll, one reveal, and one question are enough to know for sure what game the roller is playing and the probability is based on the game he is playing.
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. 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.
Is there a 1/6 camp?
It seems to me like there is a “1/11 answer is the only way to interpret the OP” camp and a “we don’t have enough information to give only one answer” camp.
Can you link to a post where someone is suggesting this. I certainly wouldn’t suggest that.
And again, this is all irrelevant fighting the hypothetical if we’re treating this as a pure math problem.
Probability problems in particular are often framed in terms of idealized, hypothetical circumstances. If a problem begins, “Thirty black balls and twenty white balls are placed in an urn, and a ball is chosen at random…” we don’t say “Nobody ever really does that,” or “How could we know that there are 30 black and 20 white balls?” or “By what mechanism is the ball chosen?” because those are irrelevant to the problem and its purpose.
@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 on the previous page for one example.
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.
IOW, my previously unstated internal position is that the 1/11 camp are probably puzzleogically right for the wrong thought-process reason.
“Puzzleogical”
Excellent.
I appreciate the portmanteau inclusion of my name.
You’ve got to pronounce it correctly: puzz-Lee-OHJ-lik-el.
sich_hinaufwinden, correct me if I’m wrong but you seem to be suggesting that what the guy would have said in the event that there were no six has some bearing on the puzzle?
I don’t see how it does, because we know that there was in fact a six. In the event of no six, he could have been planning to play Yankee Doodle Dandy on the bouzouki for all I care. It doesn’t matter, because it didn’t happen.
What did happen was that at least one six was rolled, so what matters is what his plan was for those cases. Always say “at least one is a six”? Answer 1/11. Sometimes say “at least one is a six”? Different answers, depending on the details of his plan.
With probability, we’re never just talking about one event. It’s always about many events, even if sometimes all the other events are just hypothetical. Yes, there was a six this time… but if we re-ran the experiment a thousand times, or a million, or a googol times, how many times would he say “there’s at least one six”, and out of those times, how many times would there be two of them? We can answer the second question: Out of all total rolls, 1/36 of them would have two sixes. But we can’t answer the first, unless we know under what conditions he says that.
Not what the roller would have said but rather what he would have done. I want to know what would have happened if there had been no 6.
If he had rolled for example a 5 and a 4, does he roll again until he gets a 6 (or some other target number) or does he choose between the 4 and 5 to reveal and ask what is the probability that other one matches it?
Because the OP could easily be a description of a particular roll in a game where the roller asks about doubles after every roll of the dice regardless of what they were, and in this particular roll he just happens to have revealed 6.
And if he asks about doubles after every roll, 1/6 of the time he will have rolled some kind of doubles and not have a choice of what number to reveal - it has to be one of the doubles whatever they happen to be.
And 5/6 of the time he won’t have rolled doubles and it doesn’t matter what die he chooses to reveal - he didn’t roll doubles in those cases.
So what I want to know is under what circumstances the roller asks a question about probability - every roll or only rolls involving a 6?
And how is the person being asked about the probability supposed to know the answer to that? Was he told the rules but we weren’t? Does he guess the rules from how a single roll played out? Is he supposed to interpret the phrase “at least one 6” in one specific way and that way only and where did he learn exactly what would be meant by that phrase?
In the end, I don’t think a single roll, single reveal, single question about doubles and the language used here to describe it is enough to know what rules are that the game is being played under.
And the rules of the game determine the answer to the probability question. Either this person is throwing out 25/36 of the rolls for one type of game or he is using all rolls for another type of game. And of course there could possibly be some other conditions for yet some other type of game.
So if we don’t know the rules then we can’t determine the probability.
If the answer hinges on how to interpret the phrase “at least” then this is more of a logic problem than a probability problem. And I don’t think it’s a very well stated logic problem if we are forced to interpret “at least one 6” to unambiguously mean that “I have to keep rolling until I get a 6 before I can ask about doubles”.
If re-rolling until a target number shows up is a requirement for the game, then that is important to the problem. So why wouldn’t it be simply and explicitly stated as such?
I think the ask about doubles every roll scenario requires less to assume given the limited information we have from the OP about how the game is being played.
I mean is there something simpler than a game where dice are rolled, and every roll a die is revealed, and every roll a question is asked?
Because that:
- Matches the events in the OP.
- Doesn’t require the assumption of an unspecified target number, which isn’t mentioned in the OP
- Doesn’t require the assumption of throwing out rolls or the potential to have thrown out rolls, which isn’t mentioned in the OP.
- And doesn’t require the assumption that the correct answer is not one of the choices we are given.
That being said, I still think that there is not enough information to say one way or another and I am not in the 1/6 only camp if it exists.