If a random number generator generated random numbers between 1 and 5 (including both) until it generated a 5, what is the chance that no number before then will have been generated more than once?

It should be clear (by the pigeonhole principle) that the five must be generated no later than fifth in the sequence if each other number is to have been generated only once.

Then either the five is generated first, second, third, fourth, or fifth.

P(1) = 1/5

P(2) = (4/5)x(1/5)

P(3) = (4/5)x(3/5)x(1/5)

P(4) = (4/5)x(3/5)x(2/5)x(1/5)

P(5) = (4/5)x(3/5)x(2/5)x(1/5)x(1/5)

Just add up the probabilities from there.

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Nice one. I think this is correct.

This looks vaguely familiar, it may have been asked elsewhere on MHF recently.

CB