[prolyx]

111111 is just as likely as any other number. However, in practice, humans are far more likely to think of 111111 than other numbers, so we exploit the difference between the probability a human guesses a number vs. the probability of rolling a number on a fair dice. "Did I just roll a sequence X or am I lying?" vs. "What is the probability I roll sequence X next?" are quite different questions, if by "lying" you mean you are not drawing the sequence from a fair, random source.

[elif]

111111 is just as likely as any other *SEQUENCE* of numbers. This is a little confusing because the "sequential" requirement is somewhat masked by the repeating sequence used by the example. however, the odds of rolling six 1's is 0.00002143347.

considering the non-sequential set of rolls 625631 you have the odds of exactly two 6's at 0.201 and also one 2 5 3 and 1 each at 0.402

0.402^4 * 0.201 = 0.00524928641, or ~244 x more likely.

[prolyx]

There are many ways of slicing and dicing things so that you group different sequences, e.g. group permutations as the same as you have done. But by "number" I was referring to a sequence of symbols drawn from a uniform distribution, so order matters. (On the website, you also choose options sequentially, anyways.)

[nnoitra]

No, it is not. On average you expect equal number of 1s and 0s. All 1s won't be "equally as likely" .

[pessimizer]

All 1s isn't equally as likely as the entire class of outcomes that are not all 1s, but you can also say that about every other outcome (as long as we're talking about distinguishable dice or a sequence.)

[nnoitra]

You are really confused about this. In general a randomly generated sequence on average will have equal 1s and 0s. Try generating 100K sequences and then count the sequences with:

1) equal 1s and 0s

2) all 1s

3) all 0s

See which one is more likely.

[icambron]

You missed the parent’s point. “Equal 1s and 0s” is not a sequence; it’s a class of sequences. So the fact that sequences with that property are more common than the specific sequence “all 1s” is true but doesn’t answer the question. 11110000 is a different sequence than 10101010. The question this thread is exploring is whether 1111111 is more likely than any other sequence, e.g. than 1111000 in particular (or insert any other sequence). And of course the answer is no.

[danparsonson]

The point the parent is making is that 1) represents a class of results rather than a single result, and any single member of that class is equally as likely as 2) or 3). Obviously the class as a whole is more likely than any other single result, but that's a different assertion.

[skykooler]

I believe what pessimizer means is that while "all 1s" is not as likely as "equal 1s and 0s", it is just as likely as any individual string of 1s and 0s - for example, 111111 is just as likely as 100110.