Probability of single digits from coin tosses

Question

Let's say that I wanted to generate 4 random numbers using a coin toss. I could toss the (unbiased) coin 4 times to generate one of 16 possible numbers (e.g. TTHH=0011=3) and just ignore any results higher than 9. Would this method be entirely impartial? or is it more probably to get a certain number?

Answer 1

If you completely start over afresh with four new coin tosses if you get a number $>9$, then this method is impartial (and will eventually succeed almost surely, the expected number of 4-rounds is less than $2$). Each o fthe $10$ used outcomes is equally likely because each of the $16$ outcomes in total is equally likely.

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If you did not throw away all previous coin tosses, then you would introduce some bias. Say, you just remove the first coin in order to hopefully get a digit with just one additional tossing. Then in such a second round, you could never produce $0,1,2,3$ any more, because the lowest sequence would be $THTT=4$ from a rejected $HTHT$. (Then again, you could at least reuse the last coin and save one toss per repeat; deeper tricks are possible)

Answer 2

This works fine. No different than if you needed numbers from 1-5 and used a six sided fair die. You would discard the 6 result.