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 Quote by Hurkyl Replace 11111111111111111111 with any 20-digit sequence -- chosen before the dice are rolled -- and the same is true.
I already noted that in a previous post.
In fact, if the sequence mentioned in the OP were to come up at a casino -- I WOULD be checking for loaded dice; and I would be justified in doing so.... DO you ever think I will?

 (what does "mixed" mean? every number appears at least once?) Your premise is not clear. If I operated according to the procedure Roll 20 dice and write down the sequence Come up some other sequence of 20 digits uniformly randomly Present both sequences to you then under the hypothesis that I present to you 11111111111111111111 and 66234441536125563152, the odds are 50% - 50% that the dice really did roll 20 1's in a row.
That is the premise of "future" roll. I do include it in the casino... It is, as you say -- 50/50; even Marilyn agrees to that.

 But if I operated according to the procedure Roll 20 dice and write down the sequence If the dice roll was not all 1's, write down 11111111111111111111, otherwise write down 66234441536125563152 Present both sequences to you. then under the hypothesis that I present to you 11111111111111111111 and 66234441536125563152, the odds are still 50% - 50% that the dice really did roll 20 1's in a row. Of course, if I presented you with 11111111111111111111 and 66234441536125563125, the odds are strictly 100% that the latter is what was actually rolled.
This is exactly what I was wondering about how you think. I don't care to judge the rightness or wrongness of your response -- I just wanted to know how *you* personally approached the problem.

 If I operated according to the procedure Roll 20 dice and write down the sequence If the dice roll was not all 1's, write down 11111111111111111111, otherwise select another sequence of 20 digits uniformly randomly Present both sequences to you. then under the hypothesis that I present to you 11111111111111111111 and 66234441536125563152, then the odds that the latter is what was actually rolled is $3.6 \cdot 10^{15}$ If I operated according to the procedure Roll 20 dice and write down the sequence Think up* some other 20-digit sequence that contains every digit at least once Present both sequences to you. then under the hypothesis that I present to you 11111111111111111111 and 66234441536125563152, the odds are strictly 100% that the former is what's rolled. *: The particular method doesn't matter, so long as it satisfies the given constraint
Which constraint is that?
A child playing dice with a friend, say a cup rolling dice game, refuses to show the roll sequence to their mate; but claims, it is '1111111111'; So the father comes over to stop the fight, and looks in the cup which was bumped. He sees a sequence of numbers and says to the other child, "it is either 1111111111' or '5248232123'; Then the father says to the less favored child, they are "both" equally likely. Now, we don't know what happened -- but it isn't about the probability of '5248232123' being rolled in the future. It's about what happened in an actual roll of the dice in a past game -- and cheating is suspected.

What would the other child do? (It's fair, he got all ones and that was perfect to win the game???), or would the child say "Marilyn, suppose you decided to roll dice; and then you told me '111111111' or '5248243123'; which would be more likely to be the true roll?" )
Obviously, one of the rolls is a lie -- for a dice can't be both; and it was already rolled as far as the child is concerned.

Clearly, the first child "COULD" have cheated. The total probability of the problem includes the number of ways a child could cheat according to *any* algorithm that is reasonably possible. (Let's ignore space aliens, although they *ARE* theoretically possible, they are as unlikely as 11111111111111111....).

The issue in my mind is that a child could have asked the question to Marilyn through their parent in a NON-ACADEMIC way; EG: The supposed asker of the question to Marilyn hasn't told us publicly how she came up with the question. I rather wonder if you will appreciate it if she does....

I just wanted to know how you personally thought through to an answer.
I'm not saying you're wrong or anything, I don't know your IQ score in comparison to Marilyn anyway. Why should I believe you aren't equals?

Peace. --Andrew.