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Maybe I am wrong but I think to get the probability of an arbitrary sequence appearing you have to construct a markov model of the sequence. And then I think it is a bunch of matirx multiplication that determines the ultimate probability. Basically you have to take a 6 by 6 matrix and take it to the 20th power. Obviously this is not required, but I think when people can't calculate the probabilities they tend to use intuition, which is not very good when it comes to probability theory.
The case of the die rolled 20 times nad trying to determine which sequecne is more likely is not one covered in most basic statistics courses. Yes you can apply the rule of statistics and get the right answer, but knowing the rules and being able to apply them are different things. Otherwise we could give people Euclids postulates one day and expect them to know all of geometry. I see a lot of people astonished by peoples answers, but how many of you could correctly determine the exact probability of each of the sequences appearing?