Finding Patterns in Binary Numbers: Probability

[daniel_i_l]

Suppose I have a random binary number of length n>3 and another binary number of length 3 which we will call the "pattern". What are the chances that the pattern will appear in the number such that the spacing between the digits in the pattern are equal? For example, if the pattern is 101 then it appears in 11001 (starting with the first digit and spacing of 1) but not in 111110000.
My first thought was to calculate the chance of finding the pattern starting with a certain digit together with a certain spacing and using that, calculate the probability of finding the pattern in one of the many configurations (configuration = starting point together with spacing). The problem is that some configurations are mutually exclusive - finding one lowers the chances of finding the other one while some are the opposite. How can I take these considerations into account?
Thanks.

 

[Valkarie]

The chances that the pattern will appear in the number such that the spacing between the digits in the pattern are equal is 1/2^(n-3), where n is the length of the random binary number. This is because for each of the (n-3) positions between the 3 digits of the pattern, there are two possibilities (i.e. 0 or 1). Therefore, the total number of possible configurations is 2^(n-3) and the chance that the pattern appears in one of those configurations is 1/2^(n-3).