AlanMendelson

Joined: Oct 5, 2011
• Posts: 5034
May 24th, 2022 at 1:07:56 AM permalink
Quote: gordonm888

Here is a different way to illustrate the logic of the two dice problem.

I understand that AlanMendelson will not agree with anything that contradicts his own idea, so this is aimed at others in the forum. I am trying to innovate on how to explain the logic of the solution of the 2-Dice problem.

Given: Two dice have been rolled in a cup, Joe's assistant has been instructed to say "at least one of the dice is two" after looking at both dice in the cup if and only if that statement is indeed true. Joe's assistant does look at both dice and then he does indeed make that statement.

Now, we blindfold our contestant Joe. Joe's assistant reaches under the cup and without looking at either of the dice, grabs one of the dice, removes it from the cup and places it in Joe's outstretched hand. Joe makes a fist around the die in his hand, so it cannot be seen. His blindfold is then removed.

Now we ask: "Joe, what is the probability that the dice that is still in the cup is a 2?"

And Joe replies: "if the die in my hand is a 2, I know the other die has a 1 in 6 chance of being a two. And therefore the odds of both dice being 2s is 1/6. I have already posted that statement a zillion times during the past week."

And, the assembled Elder Council of WOV Math Geeks chime in and say : "We agree with every part of that statement."

"BUT" the Wizard asks "What if the die in your hand is not a 2? What, then is the chance that the other die in the cup is a 2?" Joe starts to answer, then stops and thinks carefully. He then responds "the chance of the other dice being a 2 is 100% -not 1/6 - because I have been told that at least one of the dice must be a 2 and so it must be the die in the cup. And, of course, I also conclude that there is ZERO probability that both dice are a 2, because in this scenario the die in my hand is not a 2 !!!!!"

The Elder Council of WOV Math Geeks gently point out "Remember that the die in your hand was randomly removed from the cup and you have no further information about it. You can't just assume that it is a 2 and that the other die has a 1/6 chance of being a 2, because you must also take into account that the die in your hand may NOT be a 2, and that therefore the other die (in the cup) has a 100% chance of being a 2."

"Because to assume that a die in your closed hand is a 2 is to assume knowledge that you were not given in the statement of the problem."

Joe says "But I want to start with the knowledge that one of the dice is a two and make this a one die problem, that is, it is a problem of whether the other six-faced die is a two."

And the Elder Council responds: 'But there is no way to actually start with a die that is a 2 without peeking at it to determine it is a 2. You must accept that any die you select has a 6/11 chance of being a 2 and a 5/11 chance of not being a 2. Therefore, the other die will, respectively have either a 1/6 chance of being a 2 or a 100% chance of being a 2."

There is a 6/11 chance that any die you pick will be a 2, and in that scenario there is a 1/6 chance that the other die will also be a 2.

Thus, 6/11 x 1/6 = 1/11.

(Note: edited the name of the imaginary participant.)

This is a Rube Goldberg answer that missed the initial question.

How about handing a single die to a 3rd grader (we don't need a fifth grader or a fourth grader) and ask them to count the sides on the die?

Length doesn't make an answer correct.
Dieter
Joined: Jul 23, 2014
• Posts: 3214
Thanks for this post from:
May 24th, 2022 at 4:52:42 AM permalink
Quote: AlanMendelson

This is a Rube Goldberg answer that missed the initial question.

How about handing a single die to a 3rd grader (we don't need a fifth grader or a fourth grader) and ask them to count the sides on the die?

Length doesn't make an answer correct.

My four year old can count sides on a single d6 and get to 11. (She is precocious and cute as a button. We're still working on counting.)

Unfortunately, counting only one dice isn't going to get to the right answer for the right reason.

There seems to be a gap between what is known and what is sought, bridged by an assumption.
May the cards fall in your favor.
billryan
Joined: Nov 2, 2009
• Posts: 13347
May 24th, 2022 at 6:27:30 AM permalink
Quote: Dieter

Quote: AlanMendelson

This is a Rube Goldberg answer that missed the initial question.

How about handing a single die to a 3rd grader (we don't need a fifth grader or a fourth grader) and ask them to count the sides on the die?

Length doesn't make an answer correct.

My four year old can count sides on a single d6 and get to 11. (She is precocious and cute as a button. We're still working on counting.)

Unfortunately, counting only one dice isn't going to get to the right answer for the right reason.

There seems to be a gap between what is known and what is sought, bridged by an assumption.

And what happens when one assumes?

You end up with making an ass out of you and me.
The difference between fiction and reality is that fiction is supposed to make sense.
billryan
Joined: Nov 2, 2009
• Posts: 13347
May 24th, 2022 at 6:30:31 AM permalink
Quote: TravisR

It's creative, but it's different situation than somebody rolling two dice, looking at them, announcing that the number "x" is showing on at least one of the dice and then asking what the probability that the other dice also shows "x".

I mean, the number "2" is a nice number and all but if you substitute in an "x" instead it makes all the confusion go away and you can then see that the odds of the other dice being "x" is 1 out of 6

Math is hard. Logic is harder.
The difference between fiction and reality is that fiction is supposed to make sense.
TravisR
Joined: May 22, 2022
• Posts: 31
May 24th, 2022 at 11:33:57 AM permalink
Your top graphic assumes that the first die to stop spinning is always going to be the "2".... ... That's going to be problematic I think
billryan
Joined: Nov 2, 2009
• Posts: 13347
Thanks for this post from:
May 24th, 2022 at 12:06:55 PM permalink
Quote: TravisR

Your top graphic assumes that the first die to stop spinning is always going to be the "2".... ... That's going to be problematic I think

What are you looking at?
The difference between fiction and reality is that fiction is supposed to make sense.
TravisR
Joined: May 22, 2022
• Posts: 31
May 24th, 2022 at 12:18:45 PM permalink
I was trying to reply to a post that had two graphics of dice combinations .... and a discussion about spinning dice.... I'm still learning how to navigate the thread so my apologies for any confusion
billryan
Joined: Nov 2, 2009
• Posts: 13347
May 24th, 2022 at 12:33:24 PM permalink
There are no spinning dice. The dice were thrown and landed and the correct and honest answer is there is at least one of your particular number. You know one of two dice is a Five. You don't know which die is the five.

You are answering a different question.
Everyone agrees that if we know the left die is a Five, then the right die will be a Five one time in six.
Go back and examine the question. Once you understand it, you won't keep answering the wrong question.
The difference between fiction and reality is that fiction is supposed to make sense.
TravisR
Joined: May 22, 2022