If you know it has been generated by a valid random number generator, then no.
But if you know there is a chance it came from something other than a valid random number generator, then you would have to classify sequences like 11111111 in a "highly suspicious" category.
The fact it’s equally likely as any other sequence means it should be very unlikely to appear in the study’s 3429 samples, and extremely unlikely to show up more than once.
Precisely. A difference only arises when comparing sets of strings. In our example, the number of elements in the set uniform strings aaaa...a of length L is equal to the number of distinct symbols N, while the number of district strings of length L is N^L. So if you ask "will the string be uniform?" (which is the only reason you thought up aaaa..a in the first place) you find the probability is exponentially small for long strings: N^(1-L). Really if you tend to choose a string for any particular reason other than chance, it can be exploited as long as the reason can be guessed (e.g. the dice rolls 31415 write out the digits of pi.).
Helps to think of it in binary, 11111111 (bin) == 255 (dec). Also helps to define the space of possible outcomes, 00000000 -> 11111111. Then ask, is this a discrete event or a sequence of discrete events? I.e. if we have a roulette wheel with 256 slots, and throw a ball in, then the chance of the ball falling in any slot is 1/256.
But what if we say, we're going to generate that binary sequence by 8 successive flips of a coin, and we are aiming for 11111111 specifically? Then we have to multiply eight times. (0.5)^8 == .00390625 == 1/256
Where it gets a bit tricky is if we ask people to place bets after each successive flip of the coin. For example, starting with no flips, ask players to bet on the likelihood of 8 heads in a row. Next round, bet on the likelihood of 7 heads in a row, knowing the first was a head, etc. What minimal odds should the house give after each flip in order to reliably turn a profit on this game? Does it matter how many players are at the table when it comes to calculating those odds?
It’s about the reference class. The class of allsame sequences has only a few members. The class of … random numbers all over ala 71833791 has a lot of members. So the probability of seeing the former class is tiny.
Think of it in terms of coin tosses. Two heads is less likely than a head and a tail, because there are two sequences, 10, and 01, that map to the class of having one tail and one head. But there is only a single sequence of 00 in the class of all heads.
You can estimate the probability distribution that generates this sequence as 0: 0, 1: 1. This is as far from 0: 0.5, 1: 0.5 (a fair coin toss) as you can get.
Comparing mean and std dev can be used to estimate the distance between two distributions. See also, statistical testing.
