In a random binary sequence, does the $\phi$ ratio come into play?

Question

I have recently been working on binary sequences trying to understand how often a $0$ or $1$ repeats in the following digit, i used a random generator and wrote down the number of patterns inside the sequence, for example$ "0 1 1 0 1 0 0 0 1 0 1 0"$ would give me a secondary set of numbers$ "1 2 1 1 3 1 1 1 1"$ i then divided to find the mean and discovered (i think) that the longer the initial sequence, the closer to the phi ratio the final result will be. For the sequence above the found mean is $12/9 = 1.333$ I have limited knowledge of complex number theory and would love an explanation of whether this is an accurate observation and if so, why. Thanks