Hi PharoahsWin and thanks for joining and for posting.

Originally Posted by PharoahsWin View Post
You are correct, Alan, this is a trick question. I want to explain why it's a trick question. It's something I think everyone is missing the point on. The question asks, "If you roll the dice, and at least one is a 2, what are the odds that BOTH dice rolled a 2?"
I agree it's a trick question. But let's go back to the actual, original question:

You have two 6-sided dice in a cup. You shake the dice, and slam the cup down onto the table, hiding the result. Your partner peeks under the cup, and tells you, truthfully, "At least one of the dice is a 2." What is the probability that both dice are showing a 2?

The "trick" is really in how you decide to solve the problem. Clearly the two sides (1/11 vs 1/6) are divided in their methodology.

The 1/6 camp looks at the simple, in-your-face condition: one die shows a two, so what are the odds that the other die will also show a 2? That's 1/6.

The 1/11 camp doesn't look at the simple, in-your-face condition and looks at all of the different combinations of dice containing a 2 and then the single combination that has 2-2 which is 1/11.

I said it early on that it was a trick question and if you want to ignore the conditions of the question the answer is 1/36.
If you accept the condition of the question with one die already showing a 2 it's 1/6.
But if you don't accept the condition of the question and start questioning it by saying things like we don't know which of the two dice is a 2 then you will choose the 1/11 answer.

I accept that out of the two dice one is showing a 2. I don't care which of the two dice it is and with one die showing a 2 the only choice for me are the six sides on the other/second die.

Indeed how you read and interpret the question will define your answer. And because interpretations can be different there can be four answers:

1. 1/36
2. 1/11
3. 1/10
4. 1/6

What I said early on was that the question was written to throw a curve ball. I think the question was written to make you look for the 1/11 answer because of this "set-up line" that preceded the question:

Along the vein of the Two Coin Puzzle,


It was the two coin puzzle that asked you to look for the combinations of two coins, without one coin being "set" so you would get HH, TT, HT, TH.

But ignoring the line Along the vein of the Two Coin Puzzle, I accept that one die is set as 2 which removes the other five faces on that die and leaves only the six faces on the second die. When I start reading comments like we don't know which of the two dice is the 2 I just throw my hands up in disbelief. For heaven's sake -- there are only two dice and 2-1=1.

Clearly both the Wizard and "miplet" when they did their videos explaining the two dice problem they did exactly opposite what the original question stated as a condition and they rotated both dice to show the different combinations. Had they left one die frozen showing a 2 and did not touch that die they would be left with only one die to rotate for an answer of 1/6.

I've asked this over and over again: why rotate the dice known to be showing a 2? The original question told you "at least one of the dice is a 2" so why change that condition? And by rotating that die to explain the 1/11 answer they corrupted the question.

So, it was the 1/11 folks who were tricked. They were tricked into not following what the question asked. Whether or not that was the intent of the author of the question I don't know. But perhaps it was unintended because as redietz who is also a writer pointed out, the question was written very poorly. Very poorly. And as I said early in the discussion, had this been part of any national standardized test like an SAT or a LSAT it would have been thrown out just as another confusing SAT question was thrown out back in the 1960s.