1 in 42.
No one’s suggesting untrue statements or semantic tricks, and the multiple possible answers do not depend on “perverse” rules for making the statement about at least one of the dice being a six. There are at least two perfectly simple, reasonable rules that have been assumed repeatedly in this thread, that give different answers, and if memory serves Chronos put forward no less than four such rules at one point.
I don’t see why all these motivational therapy issues keep coming up. To me, the problem is quite straightforward.
An honest examiner walks from room to room, looking at pairs of randomly rolled dice on a table in each room, which he has been assured are fair dice. He sees one in which at least one is a six. That satisfies the requiremen to move on to the next step, so he comes out of the room and announces that much (only) about what he has seen. He asks me what is the probability that there is a pair of sixes on the table. I calculate the odds an tell him. What else do you need to know, in order to be assured that you have sufficient information?
Another honest examiner walks from room to room, looking at pairs of randomly rolled dice on a table in each room, which he has been assured are fair dice. At some point (maybe after some time limit) he is required to stop, randomly choose one of the dice in the current room, and announce “at least one of the dice was a <x>”. He happens to say “at least one of the dice was a six.”
How do you know which examiner it is? The behaviour of each examiner looks identical to you.
(my highlighting, of course.)
That red bit is an assumption that isn’t reasonable to make. (Unlike the assumptions that the examiner is honest or that they are fair (six-sided) dice, to give some examples.)
Interestingly (and as has already been pointed out), if we trust the question setter, we’d have to reject your interpretation because 1/11 wasn’t one of the multi-choice answers given.
I agree! It’s fairly clear that this was the puzzle intended; and some of the alternate protocols proposed are far-fetched.
I think the people showing alternate protocols should avoid these complaints by incanting something like “Strictly speaking, the puzzle should be written less ambiguously (as jtur88 has done above) but …”
What doubles the confusion is that some Dopers are computing the wrong probability for the model they select. Thus there are two intertwined subthreads: a vehement argument over where the problem statement is crystal clear or ambiguous (which belongs in IMHO or Mundane-Pointless I think, not GQ), and separate disagreements involving probability arithmetic. (When someone writes “1/6” it isn’t clear whether they’re using an alternate model, miscalculating for the normal model, or just making a typo.)
~ ~ ~ ~ ~ ~ ~
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]
I can’t believe that the algorithm defined in the first post could yield 1:6 after ten million iterations if it were coded correctly (it should yield 1/11 (or rather an approximation of 1/11)).
Perhaps you messed up in step 3. There you should be counting the instances with 2sixes showing.
Did you “re-roll” the “second” die in step 3?
Even with that possible suggestion made, I don’t think it would make a difference since it’s additional information we weren’t privy to in the question.
Example: 1,000,000 lottery tickets are sold in NJ, one of which is a winner. I buy 1 ticket. I believe my probability of winning is 1/1,000,000.
A friend tells me I’m silly for believing that because he has perfectly reliable inside information that the winning ticket will be/was sold in Ocean County, and he knows that I didn’t buy mine anywhere near there. He’s also aware that I didn’t have that inside information until after he told me I was silly. Was I wrong in my initial belief?
Hey, Empirical result of random event: Statistics and proof just got rolling, as it were, using the same post you are replying to as its premise. http://boards.straightdope.com/sdmb/showthread.php?t=806837
In fact, I saw your thread, and it was in that thread I saw Musicat state that he found 1/6 using a Monte-Carlo simulation. So I searched here for his algorithm. It turns out he coded something else.
Not silly. You made a reasonable assumption about something that was not explicitly stated.
The question in this thread is whether it’s reasonable to assume that our anonymous dice commentator was **obliged **to say that at least one die was a six, or merely happened to do so for other reasons. I think it’s a less reasonable assumption, and indeed various plausible other reasons have been suggested.
I didn’t make an assumption in my lottery example. I calculated probability based on the information given.
Look again at the post by ftg I was responding to:
To think that the person rolling the dice would only make the statement “At least one is a 6.” under restricted circumstances such as only when the dice were a 6 and a 3 is perverse.
Someone suggesting such a thing is adding more information, just as my friend was in the lottery example. Whether or not the winning ticket was/will be sold in Ocean County is irrelevant, just as is the person rolling the dice would only make the statement “At least one is a 6.” under restricted circumstances such as only when the dice were a 6 and a 3. It’s information we weren’t given in either probability question, so telling people that they didn’t calculate a probability question correctly because with additional information a different result would be yielded is idiotic.
Simplest is to treat probability as dependent on, and directly related to, ignorance.
For example, suppose you are playing Craps; the come-out shake is 6 (Five and Ace) but you can’t necessarily see the dice. Your best estimate of the probability of Pass Line success is
[ul]
[li] 49.3% if you see neither die;[/li][li] 45.5% if you see both dice;[/li][li] 60.7% if you glimpse a single die, which reads Five;[/li][li] 36.5% if you glimpse a single die, which reads Ace;[/li][li] 63.2% If a robot set for Five detection informs you there is at least one Five;[/li][li] 39.8% If a robot set for Ace detection informs you there is at least one Ace;[/li][li] 0 or 100% if you are a time traveler and have already seen the outcome.[/li][/ul]
There absolutely *are *situations where added information does alter the probabilities. Your lottery example is yet another case where the answer you get depends on the question being asked.
Your calculation of *a priori *1/1M odds wasn’t wrong *at the time of calculation *absent any other info. If you later learned for a fact that the winning ticket would or had been sold elsewhere then the winning probability of the ticket you now hold has changed. Or more precisely, *your knowledge *about the probability has changed.
Talking about “the probability” and “knowledge about the probability” are two different things. Sloppy language and sloppy thinking can both lead to people talking past each other on this issue.
If I have just rolled a fair die inside a sealed opaque box what are the odds I rolled a three? My knowledge is in a state of uncertainty so the knowledge odds are 1/6. But the die has already stopped moving and is either a three where my actual odds are 1 or it’s not a three where my actual odds are 0.
There is no paradox here because the two totally factual but utterly different answers are answering two different questions.
Just about everyone in the thread, myself included, has been guilty at least once of failing to be explicit on this point. Some folks have seemed to not even realize the issue exists. Those folks are unambiguously wrong.
Late last sentence add: Or at least certainly incomplete. Unwittingly so.
You assumed that the winning ticket could be one sold anywhere, not just one sold in Ocean County. Entirely reasonable, at the time. Unless the NJ lottery is corrupt, it could have been.
As for adding information, the people who are arguing for 1/11 or 1/6 or any one answer are doing just that. In the case of the 1/11 camp, they are adding the information that the dice observer was obliged to say that at least one dice was a six, should that be the case. That’s not stated in the OP, nor implied.
We may (or may not) also be brushing up against different interpretations of what probability actually means. I’ve read just enough to know that there are different philosophies or theories or interpretations of probability, and that there’s some controversy over whether it makes sense to talk about the probability of a single isolated event, but not enough to really understand or explain the different positions.
Nope. Didn’t assume anything. Not even assuming that the lottery commission isn’t corrupt and planned what store it will sold at. Doesn’t matter. My probability was still 1/1,000,000 based on the information I had.
Again, keep this to the post I was responding to:
To think that the person rolling the dice would only make the statement “At least one is a 6.” under restricted circumstances such as only when the dice were a 6 and a 3 is perverse.
It doesn’t matter if that’s a possibility. Just as it doesn’t matter if the lottery commission turned out to be corrupt. We work with the information we have. Without knowing that the dice observer makes statements under restricted circumstances, we treat the observer as he doesn’t. Hell, even if we know that he does make statements under specific circumstances, it still doesn’t matter unless we are told what that circumstance is. There are an infinite amount of specific circumstances, so we treat it as if he made the statement randomly, just as we treat the distribution of lottery tickets.
Exactly. Good link for the folks needing some more info.
Well, in that case your result is unjustified. The correct answer would be “the probabilty cannot be determined.”
Anyone saying that the OP’s puzzle has one particular answer such as 1/11 is indeed assuming that “the dice observer makes statements under restricted circumstances.” That is the very problem. They are not “treating the observer as if he doesn’t.” They are including “restricted circumstances” that are not stated or suggested in the puzzle.