looking into a inf sequence through a window - the prob of a fixed length sequence

Let's imaging there is a generated sequence of 0 and 1 of infinite length:

i = 1, 2, 3, ...

____ |<- Xi ->|
...... 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 .......
______________|<-Yi ->|

Xi = 6, Yi = 5, X_i+1 = ...

and the length of consecutive 0s is denoted as Xi and the length of consecutive 1s is denoted as Yi.

Assume Xi and Yi are generated random number following exponential distribution, i.e.
p(x) = 1/a * exp[-(x-1)/a], x=1,2,3...
p(y) = 1/b * exp[-(y-1)/b], y=1,2,3...


now we group the sequence into consecutive of windows with length W(integer).

For example, say W = 6, we may have

...... 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 .......
____|<- W_i -->|<- W_i+1->|

W_i = [0 0 0 0 0 0], W_i+1 = [1 1 1 1 1 0],
and of course there are 2^W different combinations.

My question is what's the occurring probability of a combination in the window? or we may start from some extreme case first, what's the probability when the window get [0 0 0 0 0 0]?

Thanks for your help!