TITLE: Why does the order matter when I throw differently colored balls into bins? (with limited bin capacity)
QUESTION [7 upvotes]: I have $n$ bins, $n$ red balls, and $n$ green balls. I throw the balls into randomly selected bins one at a time. Once a bin has $2$ balls in it, I close the bin. It matters to me how many bins will have two red balls at the end.
Does the order I throw the balls matter? 
The answer is yes, which I've verified through simulation and looking at the $n=2$ case, but I'm wondering whether there is a powerful conceptual explanation. This could take the form of calculating the expected number of red/red bins for any $n$ in ordered vs randomized sets of balls and noting the difference in the process, or it could take some other form.

REPLY [2 votes]: A simpler way to see that the order of balls matters in a stronger sense than Henning Makholm's answer is the following.
Start by throwing $n-1$ red balls into bins. Something happens: $a$ bins have two red balls, $b$ bins have exactly one, and $c$ bins are empty. We'll condition on the values of $a,b,c$. Notice that $c\geq 1$. If $b=0$ then what happens next can't affect the outcome, so assume $b\geq 1$ (for any $n\geq2$, this has positive probability).
Now consider two cases. If the next ball is red, the chance it completes a red/red bin is $\frac{b}{b+c}$. If the next ball is green and the ball after that is red, the chance that the red ball completes a red/red bin is
$$\frac{b}{b+c}\times\frac{b-1}{b+c-1}+\frac{c}{b+c}\times\frac{b}{b+c}=\frac{b}{b+c}\times\left(\frac{c}{b+c}+\frac{b-1}{b+c-1}\right).$$
Note that
$$\frac{c}{b+c}+\frac{b-1}{b+c-1}=\frac{c(b+c-1)+(b-1)(b+c)}{(b+c)(b+c-1)}=1-\frac{c}{(b+c)(b+c-1)}<1.$$
Therefore the probability of completing a red/red bin with the final red ball, given what happened so far, is strictly less if it is in position $n+1$ than position $n$, for all situations where this event is possible. 
It follows by averaging that the orderings R...RGRG...G and R...RRGG...G give different expected numbers of red/red bins for every $n\geq2$.