The next 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!
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!”
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.
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.
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 Haldursonet 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!
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?”
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:
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:
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.
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.
