Halt with 50% non-Halting
Perl
fork || do {sleep(1) while(1)}
Each time you run this program, it halts and doesn't halt.
Python
import random
p=.3078458
while random.random()>=p:p/=2
Each time around the loop it breaks with exponentially decreasing probability. The chance of never breaking is the product \$(1-p)(1-\frac{p}{2})(1-\frac{p}{4})...\$ which is \$\frac{1}{2}\$. (Obligatory comment about floating point not being exact.)
JavaScript
Alternatives halting and not halting. (halts on first run, doesn't halt on second, ...)
var h = localStorage.halt;
while (h) localStorage.halt = false;
localStorage.halt = true;