It was two years ago today...

[Alan Mendelson]

You don't need to have dice, let alone rotate them, to figure out the answer.

If you did have two dice and you didn't rotate them (because you cant rotate dice after they come to rest) you'd see you couldn't have an answer of 1/11.

If you want the answer of 1/11 I gave you the question that would yield that answer.

[mickeycrimm]

Let's color code the dice. One die is RED and the other is GREEN.

Combinations with at least a two in them:

Red 1, Green 2
Red 2, Green 2
Red 3, Green 2
Red 4, Green 2
Red 5, Green 2
Red 6, Green 2
Red 2, Green 1
Red 2, Green 3
Red 2, Green 4
Red 2, Green 5
Red 2, Green 6

Problem #1.

Our friend looks under the cup and says "At least one die is a two." What is the probability that both dice are 2's?

Why is the answer 1 in 11 and not 1 in 6? Because we don't have enough information to determine it is 1 in 6. All we know is that one of the dice is a 2. With that limited information all we can do is calculate the number of combinations that have a 2 in them. There are 11 and they are listed above. There is only one combination where both the dice are 2's. With the limited information we have we cannot eliminate any of the 11 combinations from possibility. So the correct answer can only be 1 in 11.

Problem #2

Our friend looks under the cup and says "The RED die is a 2." What is the probability that both dice are 2's?

What can we do with the information given here? Well, we can do a lot more with it than we can with the information given in problem #1. We can eliminate all the combinations that don't have a RED 2 in them. They are:

Red 1, Green 2
Red 3, Green 2
Red,4, Green 2
Red 5, Green 2
Red 6, Green 2

That leaves only six possible combinations. In one of those combinations both the dice are 2's. So the correct answer can only be 1 in 6.

Problem #3

Our friend looks under the cup and says "The GREEN die is a 2." We can do the same here and eliminate the combinations that don't have a GREEN 2 in them. So the correct answer here is also 1 in 6.

That's the best I can do in explaining it, folks.

[Alan Mendelson]

Let's color code the dice. One die is RED and the other is GREEN.

Combinations with at least a two in them:

Red 1, Green 2
Red 2, Green 2
Red 3, Green 2
Red 4, Green 2
Red 5, Green 2
Red 6, Green 2
Red 2, Green 1
Red 2, Green 3
Red 2, Green 4
Red 2, Green 5
Red 2, Green 6

Problem #1.

Our friend looks under the cup and says "At least one die is a two." What is the probability that both dice are 2's?

Why is the answer 1 in 11 and not 1 in 6? Because we don't have enough information to determine it is 1 in 6. All we know is that one of the dice is a 2.

You still don't get it, do you?

That's all the information you need to know to figure the answer is 1/6. There are only two dice. If at least one is a six you know that 2 dice - 1 die = 1 die and one die has six faces on it, one of which is a 2. Is that really so difficult?

Really, is it?

Regnis gets it. Maybe he can explain it better than I do.

[mickeycrimm]

Alan, the only information you have to make your determination is that at least one of the dice is a two. There are 11 possible combinations with "at least a two" in them. Could you show us from the list below which five combinations you eliminated from possibility to get to the answer of 1 in 6.

Combinations with at least a two in them:

Red 1, Green 2
Red 2, Green 2
Red 3, Green 2
Red 4, Green 2
Red 5, Green 2
Red 6, Green 2
Red 2, Green 1
Red 2, Green 3
Red 2, Green 4
Red 2, Green 5
Red 2, Green 6

[Alan Mendelson]

No Mickey. If one of the dice is showing a two, then the other die has one out of six sides. It's that simple.

Why are you making it complicated? Why are you considering eleven possible combinations with a 2?

We know that one die can be excluded. Why are you including it again?

[mickeycrimm]

No Mickey. If one of the dice is showing a two, then the other die has one out of six sides. It's that simple.

Why are you making it complicated? Why are you considering eleven possible combinations with a 2?

We know that one die can be excluded. Why are you including it again?

So 5 combinations just disappeared into thin air? Dice are pretty tricky, Alan. Which die is showing the 2? The red die or the green die? That is the crux of the problem. You don't know which die is showing the two. So none of the 11 possible combinations can be eliminated.

[Alan Mendelson]

Mickey if you want the answer to be 1/11 CHANGE THE QUESTION.

Ask: given all of the combinations of two dice that include the number 2, how many combinations can be 2-2?

That answer is 1/11.

My question is the correct question to get the answer 1/11.

You guys are not answering the question that was asked.

[Alan Mendelson]

Did you ever play Jeopardy? They give you the answer and the contestants have to come up with the correct question.

[Alan Mendelson]

Which die is showing the 2? The red die or the green die? That is the crux of the problem. You don't know which die is showing the two. So none of the 11 possible combinations can be eliminated.

It doesn't matter which die is showing a two.

Let me repeat that: given the problem, that there are two dice in the cup, it doesn't matter which of the two dice in the cup is showing a 2. This is why the dice are in a cup.

[mickeycrimm]

Which die is showing the 2? The red die or the green die? That is the crux of the problem. You don't know which die is showing the two. So none of the 11 possible combinations can be eliminated.

It doesn't matter which die is showing a two. Let me repeat that: given the problem, that there are two dice in the cup, it doesn't matter which of the two dice in the cup is showing a 2. This is why the dice are in a cup.

As I said, Alan, dice are tricky. I know I'm not going to change your mind. I'm not even trying to. This thread has had 1800 views in just four days. That means a lot of people are watching it, not just those who have participated in the discussion. I've put up the best argument I can for 1 in 11. It's for their benefit. They need to know there is a counter argument to your argument that it is 1 in 6. They've been given information by you. And they have been given information by several of us in the 1 in 11 crowd. They'll make up their own minds about it. The world won't quit turning no matter what they decide. Let the chips fall where they may.

[regnis]

Here are 2 dice. One shows a 2. What are the odds the other is a 2?

I don't know--third base

The answer is 0, because the effin dice don't flip and the other die is a 1. The point being that once one die is a 2, that can't change. You have the necessary information to determine that the odds of the other die showing a 2 would be 1/6, except in my example.




Answer to follow later

[regnis]

Here are 2 dice. One shows a 2. What are the odds the other is a 2?

I don't know--third base





Answer to follow later

The answer is 0, because the effin dice don't flip and the other die is a 1. The point being that once one die is a 2, that can't change. You have the necessary information to determine that the odds of the other die showing a 2 would be 1/6, except in my example.

[Alan Mendelson]

The answer is 0, because the effin dice don't flip and the other die is a 1. The point being that once one die is a 2, that can't change. You have the necessary information to determine that the odds of the other die showing a 2 would be 1/6, except in my example.

They'll never understand because somewhere along the line someone told them that whenever you have a dice problem you have to look at all of the combinations on the dice. Never were they told to use common sense or to consider the conditions of the problem and in this case the condition of the problem is that there are two dice in a cup and only two dice and the dice don't move after they are slammed down on the table.

You do not consider the various combinations of dice that include a two. You consider only the problem in front of you and that the observer TRUTHFULLY told you that AT LEAST one of the dice is showing a two.

How friggin hard is it? I guess for people who can't interpret the question it is very hard.

[mickeycrimm]

A few pages back in this thread I asked Alan to ask the craps dealers how many combinations add to 10 when you are rolling three dice. Alan quickly wrote it off as "bullshit questions about three dice." As Alan has said, a die has six sides. But the fact that it has six sides has led some people to draw some wrong conclusions about dice. The question of how many combinations add to 10 when you roll three dice played a roll in the history of the study of the probabilities of dice throws. It's the story about the dice gambler and the dice hustler.

In very early 17th century Italy a dice gambler got introduced to a new gambling game played with three dice. I call the person that introduced him to the game the "dice hustler." It didn't matter who rolled the dice. They were betting on totals. And the only two totals that mattered were the nine and ten. All other outcomes were no action.

The dice gambler bet the nine and the dice hustler bet the ten. When the total added to nine the hustler paid the gambler. When the total added to ten the gambler paid the hustler. The dice gambler had streaks where he did pretty good....but over time he was slowly but surely losing all his money to the dice hustler. He couldn't figure it out. He thought he had the best number. He thought he should be winning. After all, the dice are six sided....three times six is eighteen....half of eighteen is nine....that's the mean number....it should be the best number to have. But he kept losing and losing and losing.

Finally he had had enough. He quit the game. But he still wanted to know why he lost. He had a friend, a noted scientist and mathematician by the name of Galileo. He took the problem to Galileo.

Galileo studied the dice and the game for a couple of days then gave his friend an answer. He told his friend that there were 216 total combinations with three dice (6X6X6). Twenty-seven of those combinations added to 10, but only twenty-five combinations added to 9.

Because of this event Galileo was credited with being the first scientist/mathematican to work out the probabilities of dice throws. But you can bet your ass the dice hustler knew the reason the dice gambler was losing long before Galileo did.

[a2a3dseddie]

How about this:

In a standard game of craps considering only the rolls where "At least one of the dice is a 2." What are the odds of being paid on a Hard 4 bet?

Let's see, you CAN'T have an "easy 4" since you know "At least one of the dice is a 2." so 1,3 or 3,1 have been ruled out.

So, your only loss comes if a 7 appears... 2,5 or 5,2

The odds of a 5 appearing on a single die is 1/6 (the same as another 2 appearing).

Do you think it's an even money bet by disregarding the possibility of a 5,2? And what wording in the original question would make you disregard the possibility of 5,2?

Or are the odds actually 2-1 against you? (2,5 or 5,2 versus 2,2)

Alan/regnis

What are were your thoughts on this previous post?

[Alan Mendelson]

How about this:

In a standard game of craps considering only the rolls where "At least one of the dice is a 2." What are the odds of being paid on a Hard 4 bet?

Let's see, you CAN'T have an "easy 4" since you know "At least one of the dice is a 2." so 1,3 or 3,1 have been ruled out.

So, your only loss comes if a 7 appears... 2,5 or 5,2

The odds of a 5 appearing on a single die is 1/6 (the same as another 2 appearing).

Do you think it's an even money bet by disregarding the possibility of a 5,2? And what wording in the original question would make you disregard the possibility of 5,2?

Or are the odds actually 2-1 against you? (2,5 or 5,2 versus 2,2)

Alan/regnis

What are were your thoughts on this previous post?

I pass.

I am not a mathematician. I can just tell you what happens when I look at two dice.

[mickeycrimm]

Here are 2 dice. One shows a 2. What are the odds the other is a 2?

I don't know--third base.Answer to follow later

The answer is 0, because the effin dice don't flip and the other die is a 1. The point being that once one die is a 2, that can't change. You have the necessary information to determine that the odds of the other die showing a 2 would be 1/6, except in my example.

Trump changed the name of fake news to very fake news. Your answer is not only wrong it is very wrong. It doesn't make you a bad person though. Just very wrong. The only information you have is that "at least one die is two." Remember, the dice are under a cup and only your friend see's them, you don't. You don't know which die is the unknown die. Not one of the 11 possible combinations that have "at least a two" in them can be ruled out. You don't have enough information to do that.

[mickeycrimm]

I am not a mathematician.

I absolutely agree. See there, we CAN agree on something.

[mickeycrimm]

Mickey if you want the answer to be 1/11 CHANGE THE QUESTION. Ask: given all of the combinations of two dice that include the number 2, how many combinations can be 2-2. That answer is 1/11. My question is the correct question to get the answer 1/11.
You guys are not answering the question that was asked.

This is the question, the original question, that I answered. "your friend looks under the cup and says "at least one of the dice is a two. What is the probability that both dice are 2's?" This is the question I answered. It is a clear and concise question. I totally understand the question. I DID NOT, repeat, DID NOT, repeat, DID NOT, change the question. And I answered it clearly and concisely. When the only information you have is that "at least one die is a two" you cannot eliminate any of the 11 possible combinations from possibility. Both the dice are 2's in only one combination. Therefore, the answer is 1 in 11.

[slingshot]

And that is why, Alan, when you finish a VP session and return another day, the machines will pick up right where you left off. The millions of hands others play will not count, so short term, goal oriented play is useless, and only AP play wins.