# Randomly choosing objects from a set of objects - guarantee all objects are picked at least once?

I got into a debate with someone in class over an experiment that was claimed to be randomized.

The experiment was such that there is a bucket of distinct objects, and objects were being "randomly" (but independently) selected from the bucket. There are $$N$$ objects in the bucket, and $$M$$ random selections are made. The experiment was conducted such that $$M>>N$$, but $$M$$ is finite.

It was claimed that this experiment (choosing objects $$M$$ times) was conducted "randomly," and also that the experiment GUARANTEED that all $$N$$ objects would be chosen at least once. I think that "randomly" and "GUARANTEED" are mutually exclusive in this instance.

I made the argument that if the experiment guarantees that all $$N$$ objects would be chosen at least once, then the selections are not conducted randomly. To guarantee that all $$N$$ objects are selected, with a finite $$M$$ (albeit being much larger than $$N$$) would remove the randomness in my opinion? Am I right, or is it possible for this to still be random while guaranteeing all objects are selected at least once?

Note that I am not asking what is the probability of selecting all $$N$$ samples by making $$M$$ selections, where $$M>>N$$. I understand the probability here is close to 1. For the probability to be equal to 1, then $$M\rightarrow\infty$$.

• You can select $M=N$ times randomly from $N$ objects and still guarantee each has been selected: just mix the objects up thoroughly and withdraw them from their population one at a time without replacement. This (standard) example shows that "random" and "guarantee" are not necessarily mutually exclusive properties. – whuber Mar 31 '19 at 15:27

There’s a distinction between asking what you’ve picked after $$M$$ draws, vs conditioning on the event that you’ve picked all $$N$$ objects after $$N$$ draws. The latter event is still random, for example, the total number of times each object was chosen will again be random.
• Sorry, I don't think I understand. They claimed the experiment GUARANTEED that each of the $N$ objects would be drawn at least once, but that the selections were performed "randomly." I don't see how an experiment that was making $M$ draws can "guarantee" that all $N$ objects are selected at least once? – anonuser01 Mar 31 '19 at 2:40
• Yeah I agree, and that was the point that I was trying to make in the OP as well as during my class debate. The person seemed to suggest that because $M>>N$, this is what "guaranteed" each object was picked at least once. I think he might have used his empirical observations (which showed that each object was drawn at least once for all the experiments they conducted) to make a generalization that the probability is 1. In reality, the probability is probably close to 1, given that $M>>N$, so he may have just been lazy with words. – anonuser01 Mar 31 '19 at 2:51