My example said that one winning lottery ticket out of a million will be sold. It’s irrelevant whether or not the store it’s sold at is determined randomly or hand-picked. Without knowing how it’s hand-picked, we’re left with my probability of buying the winning ticket at 1/1,000,000.
And once again, look at the quote by ftg I am responding to.
Re ftg’s quote, if the observer’s rule is “say that at least one dice is a six only in the cases 3-6 and 6-3” (I think that’s what **ftg **means), and the dice are rolled and the observer does indeed say that, then the probability that the roll was 6-6 is obviously zero.
It seems from your reply that you are saying the probability in that case is something else?
Oh wait, you’re saying he uses that rule, but you don’t know it. Hence the answer remains <something definite>. Same as the lottery thing.
And as with the lottery thing, I say “no , the probability cannot be determined, because you don’t know his rule, and never knew it in the first place.” There never was a definite answer. Not without making some unjustified assumptions.
So the question is “What are the odds that full disclosure will eventually yield a 6?”, and I don’t have a problem with that. Probability is defined as exactly that – The mathematical likelihood of an outcome, when viewed in an intermediate chronicity between what’s been (albeit imperfectly) disclosed cause and the undisclosed effect.
I know a billionaire that is sponsoring his own lottery, Cliffy. He sold only four lottery tickets for $1 a piece. Each ticket was sold at one of four stores in my town. There is one winning ticket out of the four, and it pays out $1,000,000. I bought one ticket. The thing is, I don’t know if he distributed the tickets randomly at these four stores or hand-picked the store the winning ticket was sold at. I bought mine at the store closest to my house. The winner hasn’t been announced yet. What is the probability that I will win? Or can the probability not be determined? If the probability can’t be determined, why?
That’s almost a good statement of the end question. The end question is actually “What are the odds that full disclosure will eventually yield a *second *6?”.
The immediately preceding question is “The observer told us something about the state of the just-rolled-and-now-stopped dice. But what did he say and what did that really mean?”
The answer to the intermediate question (along with fair, two dice, both 6-sided, and truthful observer) *determines * the answer to the end question. Absent an answer to the intermediate question no definite answer can be given to the end question.
So please answer the intermediate question for me. Given that answer I can unambiguously, correctly and trivially compute the relevant odds to answer the end question. No guessing allowed in your answer though.
Also, many would disagree with your definition of probability. What you suggested isn’t wrong, but it is only a piece of one aspect of one definition amongst several competing definitions. Ref Thudlow Boink’s link above for more.
Strictly speaking it cannot.
Making the supporting assumption of random distribution your odds go from indeterminate to 1:4. Making other supporting assumptions yield different answers.
For a real government-run lottery making the assumption of random distribution of tickets (or a fair draw of ping pong balls) is pretty well-founded albeit not perfectly so. Conversely for some Mob numbers racket or your example lottery the assumption becomes less plausible. Perhaps yuugely less so if run by certain celebrity fat cats.
To the degree we have data to quantify the degree of “less plausible”, the odds become increasing calculable with declining error bars as the quality of our data goes up.
I’m struggling a bit to see which pin-head your angels are dancing on. You’ve got a bee under your bonnet about something. But repeatedly referring back to a single sentence in a post 6 pages ago leaves me confused. I’m not asserting here that you’re wrong. Just that you’re not communicating successfully to me, for whatever small amount that may matter to you. Others may be equally baffled.
Sorry for the long post but I’ve kept mine short till now.
Agreed.
Kind of agree but I’d like to add my thoughts on the italics part - apologies if I’m just misunderstanding you.
I don’t think merely being obliged to say there’s a six whenever one comes up is what makes it 1/11. They could have a rule where they are obliged to reveal the 6 and the probability would be 1/6, if they are asking some question after every roll.
I think what changes the probability to 1/11 is the (assumption of) specifying any “target” number before playing which means that all rolls not involving at least one of that number are thrown out. So the only rolls under consideration are the 11 combinations that have been listed many times - the ten rolls with one of the target number and the roll of doubles with the target number.
However, when the game is played such that after every roll the question about probability is asked, the odds are 1/6. And I don’t think it matters what the rule would be for picking which of the dice to show. You could always pick the red one, always pick the green one, pick one at random, pick the one closer to you, and so on, including the part of your post I am addressing which is always pick the 6 if at least one 6 comes up.
When there is no target number and a question is asked after every roll, regardless of the rule for which die is revealed, the problem really boils down to “Did I just roll doubles (of any type)”.
1st roll is 4 and 5, say “At least one is a 5”, prove it by showing you the 5 and ask the probability of the other die also being a 5.
2nd roll is 3 and 1, say “At least one is a 3”, prove it by showing you the 3 and ask the probability of the other die also being a 3.
3rd roll is 2 and 2, say “At least one is a 2”, prove it by showing you one of the 2s and ask the probability of the other die also being a 2.
4th roll (which could be the one described in the OP) includes a 6, for whatever reason or whatever rule say “At least one is a 6”, prove it by showing you the 6 and ask the probability of the other die also being a 6.
I believe the probability in each of these examples is 1/6, because once again the question is basically “Did I just roll (any kind of) doubles”.
It’s only by assuming that there is some unstated rule in which only rolls including a pre-specified 6 (or some other target number) will be considered and continue on to asking about probability that the set is reduced to 11 possibilities with one of those being doubles.
And BTW, I think the probability would also be 1/11 if a new target number was specified before every roll and only rolls with the current target number move on to the question about probability, such as:
1st roll I specify 2 as the target number, roll 3 and 4, and throw that roll out without asking about probability.
2nd roll I specify 4 as the target number, roll 1 and 5, and throw that roll out without asking about probability.
3rd roll I specify 6 as the target number, roll 6 and any other number which may or may not be another 6, say “At least one of these is a 6”, prove it by showing you a 6 and ask what the probability is that the one is also a 6.
In this scenario, whenever the roll includes the target number (which is changing from roll to roll) and the question is asked, the probability is 1/11 because we decided to only consider special sets of rolls in which the chances of doubles is 1/11.
These two sentences are in opposition to each other. You seem to think that the roller will never lie, and say that there’s a 6 present when there isn’t. I agree; that’s a perfectly reasonable assumption for a problem of this sort. But that assumption is a restricted circumstance under which the roller would say “at least one die is a 6”! If you posit that the roller always tells the truth, then you cannot posit that he’s unrestricted in what he says. He definitely has restrictions, there are multiple possible restrictions that he might have, more than one of those possible restrictions is quite reasonable, and those different reasonable restrictions yield different answers.
Let’s try one more scenario. Someone rolls two dice, looks at them, and then says absolutely nothing. In this case, what is the probability that the roll is two 6s?
The same as here:
Someone rolls two dice. What is the probability that the roll is two 6s?
But if one uses the same assumptions one uses to get 1/11 for the original problem, then the probability must be 0, because the roller didn’t say “There is at least one six”.
You lost me there.
100%. The boxcars reminded him of a backgammon game long ago on Sutter Street, final-game of a big-money Calcutta. (The man from L.A. did roll double sixes when he needed them.)
Mr. Wong remains rather loquacious. Only the sight of that disappointing 6-6 pair could render him speechless.
HTH. 
In other words, we’ve gotten to the incredibly absurd situation where someone merely says
“I just rolled one die. What are the chances it’s a six?”
And there are apparently people here who think that 1/6 is NOT the correct answer. That there are different interpretations. That the statement is ambiguous. Etc.
That’s absurd pseudo-philosophical bull. And it isn’t statistics.
Assumptions are critical to make it through most scenarios in life correct? If you took the SAT or ACT how did you make it past the first question?
You didn’t say it was a fair die. You didn’t list what was on each side. You didn’t state how many sides the die had.
Yes. But it’s important that the assumptions are justified, hence my repeated use of that word and its opposite, which you may have noticed.
The point, I think, is that the assumptions necessary to support arguments for 1/11 necessarily lead to such seeming absurdities. Which should make one question those assumptions.
The assumptions that lead to 1/11 are reasonable, but so are the assumptions that lead to other answers. A proper response, therefore, is to either ask what assumptions the questioner intended, or to specify the answers for multiple assumptions.
We’re overlooking a crucial point, I think. There doesn’t even have to be a pair of dice. Only a person who says “I rolled two dice and one of them was a 6” The odds given by the respondent would be exactly the same, whether or not there was ever an actual dice roll. Of, for that matter, ever a person who says he rolled them. (He might have dreamt it.)
Does that change the odds?