Pylongolf2, 12 bytes

1r0=?t¿1>~<

I would've saved 2 bytes if it didn't have to print 1 forever:

1           push 1 to the stack
 r          pick a random number between 0 and 1
  0=        push 0 to the stack and compare them
    ?t¿     if true, terminate, if not then skip this
       1>~< push 1 to the stack then print it forever.