Puzzle #156: Cheap, Digital LEDs

You just bought some digital LEDs for really cheap. The manufacturer states that one of the seven segments might not light up. Assuming you do not want confusion between two digits, you should use only those LEDs where the failing segment is which one(s)?

Another batch has a segment that is always on. Which segment will cause the least trouble and why?

(In your answer, number the segments from top-to-bottom, left-to-right so 1: top, 2: top left, 3: top right, 4: middle, 5: bottom left, 6: bottom right, 7: bottom.)

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, June 1, 2016 at noon Pacific Time. I will post the answer shortly after.