The Wizard will bank this bet: 1/6 vs 1/11

[OnceDear]

Let me ask the question differently: if instead of using the word probability and we used the phrase "what are the chances that the other die is also a 2" would the 1/11 answer still hold?

I'll answer, but please Alan, try to comprehend the answer FULLY.

We are NOT EVER asked what the chances or PROBABILITY is of the other die being a two. We ARE asked IN THE ABSENCE OF KNOWING WHICH DIE IS A TWO, what the probability is that we have a pair of twos. That is different. WAYYYYY different.

You need to get your head around this rather odd set of facts.

For each die, the odds of that die being a deuce would be 1/6. You can use the words odds, chance and probability without too much disagreement.

You, not unreasonably, assert that the odds of 'the other die being a two' is 1/6

Are you happy to agree with me on that? It's not really what's being argued. Really. it's not.

BUT. GREAT BIG BUT....
That is not the question.
We are NOT asked what the probability of one other die being a deuce. You adamantly believe that's what is asked. But the question is absolutely explicit. We are asked what the probability is of A PAIR of Deuces! YOU can NOT extract one die, the die that is a deuce, because only the peeker knows which die that is, and he is not telling.

Your assertion would only follow from the original question if we could identify which die is already a deuce.

We know only one thing and that 'AT LEAST one of the dice is a deuce'.

Until the cup is lifted, we cannot ELIMINATE any die. We cannot eliminate any face on any die. All we can eliminate is outcomes of dice where there is no deuce. All we can eliminate is...
1-1
1-3
1-4
1-5
1-6
3-1
3-3
3-4
3-5
3-6
4-1
4-3
4-4
4-5
4-6
5-1
5-3
5-4
5-5
5-6
6-1
6-3
6-4
6-5
6-6

UNTIL we lift the cup, none of the above outcomes could sit under the cup, but there are still those pesky 11 other ways that the dice under the cup COULD have come to rest:-
1-2
2-1
3-2
2-3
4-2
2-4
5-2
2-5
6-2
2-6
2-2

UNTIL the cup is lifted. That is the sum total of all that we know. Having a red die and a blue die, or a left die and a right die or a big die and a small die can all help us to visualise that.

If we have a big die and a little die, you absolutely cannot rule out which one is a deuce. The peeker could put it in his pocket, but until you see it, you are none the wiser.

Once you see the cup lifted, probabilities are no longer at issue: bets are off. But until then..... 1 of 11 outcomes is hiding under that cup and you and I don't know which it is. THAT is the very nature of 'chance' or 'probability' or 'odds'. the reason no-one shows you this in a video is because once the cup is lifted, we can only see one fixed outcome of both dice. You, yourself rotated the 'other' die in some effort to show the six ways it could have landed. But you would have needed to show how both dice could have landed and for that you would have to acknowledge that until the cup is lifted, you don't know which one to rotate. Your video would have had to show you rotating both dice.

By 'setting' one specific die as a deuce, you would be imposing a condition that is not imposed.

I don't know much about craps, but would the casino let you set a die and roll just the other? Well this is why, because it is breaking the rules and the payout table would be shot to hell.

Now, the more serious question. Why has no-one taken up wagers offered by Wizard, myself and others.

Alan seems to assert that the bet offered DOES model the question appropriately.
Alan does agree that our peeker saw both dice and would only declare deuces or no deuces. It's agreed that that was an original omission from the question, but this thread is very specific.

With agreement on the peeking and declaring rules, we have a pretty darned explicit question. It has only one correct answer, unlike the very original question in a distant memory far far away.

Why is not one person taking the offered wager. It's either going to favour the player or the offerer and after a few hundred rolls, variance would become insignificant. If you want more confidence, stake $5 per resolved bet over 100,000 bets. The money is there for the taking. Maybe some 'maths guy' could tell you what you could expect to win or lose and even within what reasonable margin of error.

You could even use my spreadsheet, It's free and it even tells you what I estimate that you would lose.

Even Rob Singer said

But, if as I suspect, we don't get to see which die has the original 2 until after the bet, then 9-1 is a punk bet because two 2's, after knowing half the outcome at that point, is an 11-1 event in this case.