In this problem, to determine the probability of picking marble we assume identical marble or distinct marble? In an urn, there are 3 red marbles and 3 green marbles. Two marbles are picked randomly. Determine the probability of 1 red and 1 green marbles are picked.
To solve it, I have 2 assumptions.
First assumptions, we consider the 3 red marbles are identical marbles and 3 green are identical marbles.
Let $R$ and $G$ denote the red marble and green marble respectively.
So we have the sample space is $S=\{RR,RG,GG\}$. Thus we have the probability is equal to $\dfrac{1}{3}$.
Second assumptions, we consider the 3 red marbles are distinct marbles and 3 green are distinct marbles.
So we have the sample space is $S=\{R_1R_2,R_1R_3,R_2R_3,R_1G_1,R_1G_2,R_1G_3, R_2G_1,R_2G_2,R_2G_3,R_3G_1,R_3G_2,R_3G_3,G_1G_2,G_1G_3,G_2G_3\}$.
$$\vert S\vert=C(6,2)=15.$$
Thus we have the probability is equal to $$\dfrac{C(3,1)+C(3,1)}{C(6,2)}=\dfrac{6}{15}=\dfrac{2}{5}.$$
My question:
Which one is true answer to solve this question? I'm very confused to solve this problem because there are more than 1 assumption to solve it, and I confused to determine which one the true answer. Generally, are we assume identical marble or we assume distinct marble?
 A: You should assume that the marbles are distinct (as they would be if you were picking from an urn in real life). For example, intuitively, if instead the urn contained 1000 green marbles and 3 red marbles, then the probability to pick one green and one red will be very small (as we will likely just pick two green marbles), and this only happens if we assume the marbles to be distinct.
By the way, there are 9 choices in your second assumption (the correct assumption) that result in one red and one green marble being picked. So the answer should be $\frac{9}{15}=\frac{3}{5}$.
A: The solution in the first assumption you make is wrong. Having three different outcomes does not mean they are equaly probable. If the sample space is $S=\{RR,RG,GG\}$, that does not imply that the case $RG$ has $1/3$ chances of happening. To make you understand, what you did there is like saying:
"The sample space for the lottery is $S=\{\text{win, lose}\}$, so the chances of me getting a win if I participate are $1/2$" (I bet you can see by yourself why this is wrong).
The easiest way to approach this problem is noticing that it doesn't matter which ball do you pick first, the chances of getting 1 red and 1 green will be the same as the ones of picking a marble of the opposite color to the one you picked first. Since there are 3 red marbles and 3 green marbles, when you take one, then there will be 3 remaining of the opposite color to te one you picked (and 5 in total), so the chances of you getting one of the opposite color are $3/5$.
