The randomness of the sequence must be more clearly defined to provide an answer.
My first thought was like the answers of Oray and gsomani. However, this assumes the following randomness rule be followed by our friend:
"Choose an element with probability equal to its remaining appearences divided by the numbers of rounds left".
However, this is still an assumption. Observe an alternative:
"Choose an element with probability 40% rock, 35% paper, 25% scissor, and normalize if any of them have consumed all their appearences". It is very arguable whether this is closer to the natural behaviour of humans, as Dmitry adds.
Those have vastly different strategies. In the case where, after a few rounds, there were 2 rocks and 10 scissors remaining, the first strategy would lead you to choose scissor (expecting it to appear with 10/12 probability), while the second one would lead you to choose rock (with probability ~60%).
After Dmitry defined the randomness as a randomized vector, I think it comes closer to the first one. However, conditional probabilities have tricked me times and again. The question is: If there is such a randomized vector, and we know there are such predetermined totals, and we know the first N results, what is the probability of each value for the N+1 result?