Confusion is possible, but we got numerical evidence against
popular belief about the normality of $\pi$ in base two.

According to [wikipedia](https://en.wikipedia.org/wiki/Normal_number)

> a real number is said to be **simply normal** in an integer base b if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b. A number is said to be **normal** in base b if, for every positive integer n, all possible strings n digits long have density b^-n. 

Working with precision ten thousands binary digits and n=2,
the counts of the strings in $\pi$ are:
$(11: 1661, 10:  2505,  01:  2505, 00:  1659)$

$10$ occurs about 1.5 times more than $11$.

$\pi$ appears to be simply normal in base four.

The same discrepancy happens for $\sqrt{2}$, $\log{3}$ and
large random integers.

Is $\pi$ not normal in base two and $n=2$?

Computations were done with sagemath and pari/gp.

**Added** The shorter of the two programs, are there obvious bugs in it?


     sage: pre=10^4
     sage: gp.default('realprecision',pre)
     0
     sage: sp=gp.binary(gp.Pi())
     sage: sp2=eval(str(sp[2]));sp3="".join(str(_) for _ in sp2)
     sage: sp3.count('11'),sp3.count('10'),sp3.count('01'),sp3.count('00')
     (5586, 8289, 8290, 5529)