Transformation of distorted random sequences produces random sequences?

My program generates what might be called switch(x) sequences.

Here's a description the algorithm that produces them:

# define a switch(x) sequence of length n^2
def switch(x):
    s = []                                     # define a bit string s:
    s.append(random.choice([True, False]))     # set its first bit randomly,
    for i in range(1, n * n):                  # but for every subsequent bit i,
        prior = s[i-1]                         # find its prior bit i-1,
        shouldSwitch = x > random.random()     # and with probability 1-x,
        if shouldSwitch: s.append(not prior)   # set i to opposite of its prior;
        else: s.append(prior)                  # otherwise, have i = its prior.
    return s

At x = .5, the sequence is thought to be a perfectly random sequence of bits. Deviating x distorts this randomness by producing sequences that either alternate or repeat bits too often.

I produced a program that computed the average alternation rate of a produced switch(x) sequence.

r = 0.0
for i in range(len(s)-1):
    if s[i] != s[i+1]:
        r = r + 1
rate = r/(len(s)-1)

Of course, I always obtain a rate relatively close to x when I put a switch(x) sequence through it, whatever x is. Like, within 1/100ths.

But suppose I transform my produced switch(x) sequence like so (where len(s) = n*n):

s1 = switch(x)
s2 = []
for i in range(n):
    for j in range(n):
        s2.append(s1[i + j*n])

Whenever I calculate the alternation rate for sequences transformed in this way, I always obtain a value really close to .5! The consistency is scary.

That doesn't make much sense to me, especially for values of x that are very close to 0 or 1. So I was hoping you could help me figure out what's going wrong.

Sorry if something's terrible about my style/efficiency.