beausoleil

Now there's irony!

There are 36 combinations of 2 dice, each equally likely.

Of these, 11 have at least one 6.

Saying 'there is at least one 6' effectively says that only these 11 combinations are under consideration.

In only one of them, 6-6, is the second dice a 6.

Hence 1 in 11.

If your argument were correct, throwing 6-6 at craps would be as likely as throwing 5-6. Care to bet?

beausoleil

Dice 1, probability 6 = p
Dice 2, probability 6 = q

Prob 1=6, 2= not 6 = p(1-q)
Prob 1- not 6, 2 = 6 = q(1-p)
Prob both 6 = pq

Prob second dice 6, given first is 6 = pq/(p+q-pq)

If p=q=1/6 ->

(1/36)/(1/6+1/6-1/36) = 1/11.

parkdalian

I interpret it as being like two rolls in a row. When we assume one is a six, the only way we can accurately assume one die is a six (since they are fair) is by it being revealed first, before the possibly non-six die. Revealing both at the same time makes no sense, as the mystery would be resolved instantly.

So we have:

Known Unknown
6 - 1
6 - 2
6 - 3
6 - 4
6 - 5
6 - 6

and throw out the last five of the combinations because the known die can't be 1 2 3 4 5.

The other interpretation could be that me, hoping for two sixes is told by someone who agrees to be a "six detector", knows the result of both, and says "at least one of them is a six".

In that case it appears to be 1 in 11.

Apologies to mathematician Beausoleil, you have much to teach me, this indeed looks like a matter of interpretation.

[ July 03, 2002: Message edited by: parkdalian ]</p>

seanie

I've never studied mathematical probabilities and I'm struggling to follow this. I'm coming at this from the angle of an occassional gambler and I can't see how the answer can be anything other than 1 in 6.

First off I can't see the relevance of whether the dice are rolled simultaneously, a microsecond apart, a day or a year apart. What possible influence does this have on the dice?

Secondly what's the relevance of what is seen when or whether anything is seen at all? What possible influence does this have on the dice?

We're talking about two dice that normally have a 1 in 6 chance of turning up a 6. But now, just because we've linked their outcomes together the odds have changed for one of the dice? Which one? The one you haven't seen yet? Because the dice you've seen rolled a 6 the one you haven't seen only had a 1 in 11 chance of doing so? A 6 was actually half as likely as any other result? How? By what physical process was this achieved? The outcome of the dice may not be independent for our purposes but from a physical perspective they are. The physical forces that determine how one dice lands aren't altered because of the outcome of another dice.

And throwing 6-6 at Craps is just as likely as 5-6 or any other combination.

I don't follow the logic of 1 in 11 so I'd appreciate more explanation.

seanie

Put it this way.

I roll two dice (simultaneously if you like).

I don't look at them.

I show dice A to Bob and he sees it's a 6. He concludes there is a 1 in 11 chance the other dice is a 6.

I show dice B to Alice and she see it's a 6. She concludes there is a 1 in 11 chance the other dice is a 6.

Which one is right? Are they both right? If not why not?

If Bob correctly tells me that B has a one in 11 chance of being a 6 and Alice correctly tells me that A has a one in 11 chance of being a 6, am I to conclude that the chances of double 6 are actually 1 in 121?

If not why not?

seanie

I see a problem. If they both tell me the chances are 1 in 11 I suppose I should deduce that the odds of a double 6 are in fact 1 in 1.

Nevertheless are Bob and Alice both right to conclude the odds are 1 in 11?

seanie

Put it another way.

Say I get 100 people to each throw 100 dice simultaneously.

I get 100 people to throw 100 dice sequentially.

I pick two of the dice entirely at random, without knowing which group they come from.

I look at one and it's a 6.

What are the odds of the other one being a 6?

Principia

Seanie,

You can conduct your own experiment.

Roll two dice and have two tables. If one of the two dice is a 6 (could be both), then score in one table. If both of the dice is a 6, then score in the other.

After, say, 100 scores (which may require quite a lot more dice throws!), divide the number of 2 6s by the number of 'at least one.' Your answer should not be too far off from 9-10%.

Then do the following experiment.

Roll two dice, of different colors (say red and white). Once again make two tables. In one table, score only if you see the white dice is a six. In another table, score only when the white dice is a 6 and the red dice is also a 6. After around 100 scores (once again, this would require a whole lot more than 100 dice throws), divide the number in the second table the first. The value you should get is approx. 15-17% (or larger than in the first experiment).

Good luck!

beausoleil

No it isn't because there are two combinations (5-6 and 6-5) as opposed to 1 (6-6). Similarly, a total of 7 is the most likely because it can be (1-6, 2-5, 3-4, 4-3, 5-2, 6-1)
---------------
The underlying principle is that probabilities change as relevant information is disclosed.

If I roll a dice and ask you to gues the number, you have 1 chance in 6 of being right.

If I roll a dice, look at it, and tell you it is either a 3 or a 4, you now have 1 chance in 2 of being right.

This problem is the same principle. The person who throws two dice gives you some information about what the outcome was - that at least one of the two dice was a 6. This limits the possible combinations that could have been thrown. Before he told you, from your point of view it could have been 1 of 36. Now there are only 11 to choose from, and in only one of those is the other dice also a 6.

beausoleil

Bob should now know that there is 1 chance in 11 of it beng a 6, from his point of view.
Alice also knows there is 1 chance in 11 of it being a 6, from her point of view.

('it' reffering to the die they don't see, in each case)

However, you and I both know that they are both 6s.

Probabilities to do with events that have already occurred and about which we have some knowledge behave differently from those predicting the outcome of events, in a sense. The chance of throwing two sixes next time you roll dice is still 1 in 36.