It is much easier to calculate the number of strings of a certain length that have no adjacent 1s, and this can be done recursively.
Let's assume we are trying to calculate the number of strings of length 6 that have no adjacent 1s, which we can define as f(6). Currently, we have:
where X can be any digit. If we assume the first digit is a 0, we end up with the following:
Since the only constraint is on adjacent 1s, the number of valid configurations of this string is exactly the same as the number of configurations of:
which can be calculated as f(5). On the other hand, if we choose the first digit as a 1, we get:
However, since we can't have two adjacent 1s, we also know that the second digit is a 0, and we get:
The number of possibilities here is the same as:
or f(4). Therefore, f(6) = f(5) + f(4). This can be generalized, since, in a string of length n, making the first digit a 0 shortens the string to n-1, and making it a 1 shortens it to n-2. The generalized formula is therefore f(n) = f(n-1) + f(n-2). This happens to be the formula for the Fibonacci sequence!
We also need our base case, and we can easily find that there are 2 possibilities for a string of length 1 (0 and 1). We can also see that there are 3 possibilities for a string of length 2 (00, 01, 10). Thus, our base case is f(1) = 2 and f(2) = 3.
From here, we can either calculate f(8) recursively by building up from f(1) and f(2) until we get to f(8), or just looking at the Fibonacci sequence itself. For our purposes, the sequence goes 2, 3, 5, 8, 13, 21, 34, 55..., so f(8) = 55.
Finally, we can remember what f(8) actually represents. f(n) is the number of strings of length n that have no adjacent 1s, and we are looking for the number of strings of length 8 that have at least one pair of adjacent 1s, so we need to calculate (TotalOutcomes - IncorrectOutcomes) / TotalOutcomes. For a string of length 8, there are 2 ^ 8 = 256 total outcomes, and we know from before that there are f(8) = 55 incorrect outcomes, so our final answer is (256 - 55) / 256 = 201 / 256, or about 0.785.