Sample Space for the following experiment Given there is a bag having 2 red balls, 3 blue and 4 blue balls. The experiment is to pick one ball, inspect the color and return it to the bag, then pick another one.
What is the sample space of this experiment?
I am not sure what is the experiment here, but I think the sample space would be {RB,RG, RR, BB,BR, BG, GB, GR, GG}. I also think that the order matters here so RB is different than BR. I really appreciate if someone can assure my understanding and analysis is correct.
Thank you
 A: The experiment is the description of the sampling process:

The experiment is to pick one ball, inspect the color and return it to the bag, then pick another one.

The sample space of the experiment is the enumeration of all possible elementary outcomes of the experiment.  Since the first pick and the second pick are distinguishable, so must be the outcomes; hence $(R, G)$ is a different outcome than $(G, R)$ because the former describes the outcome where a red ball is drawn first, then a green, but the latter describes the outcome where the green is drawn first, followed by a red.  So the full sample space can be written
$$\{(R, B), (R, G), (R, R), \\ (B, B), (B, R), (B, G), \\(G, B), (G, R), (G, G) \}.$$
Also notice that in our sample space, each elementary outcome is disjoint from every other outcome (so for instance, it is not possible to simultaneously observe $(B,G)$ and $(B,B)$--if one of these is observed, that means none of the others happened); and that all possible outcomes are described.
For further understanding, what would be an experimental design for which the sample space does not consider the order of the outcomes for each draw?  Obviously, one such experiment would be if you drew two balls from the bag simultaneously, without replacement.  Then there is no order to the draws.  How would you write the sample space?
Finally, here's another experiment to consider.  Suppose you have a fair six-sided die numbered from $1$ to $6$, and a fair coin.  You roll the die, and if the number rolled is a prime ($1$ is not a prime number), then you flip the coin as many times as the number you rolled on the die, and observe the outcome of each flip.  If the number you rolled is not prime, then you flip the coin once, and observe the outcome of that flip.  What is the sample space of this experiment?
