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

According to wikipedia

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.

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

According to wikipedia

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)