2 deleted 24 characters in body
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Pylongolf2, 1211 bytes

1r0=1r1=?t¿1>~<1>~<¿

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

1           push 1 to the stack
 r          pick a random number between 0 and 1
  0=        push 01 to the stack and compare them
    ?t¿   1>~<¿  if true, terminate, ifpush not1 thento skipthe this
stack then print it forever.
   1>~< push 1 to the stack then print it forever.terminate

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.

Pylongolf2, 11 bytes

1r1=?1>~<¿

Explanations:

1           push 1 to the stack
 r          pick a random number between 0 and 1
  0=        push 1 to the stack and compare them
    ?1>~<¿  if true, push 1 to the stack then print it forever.
            terminate
1
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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.