I have a jar containing `n` numbered marbles, where `1...x` marbles are red and marbles `x+1...n` are black, and I want to remove them one by one. However, red marbles are larger and more likely to be grabbed: at any time, each red marble is `k` times more likely to be selected than each black marble, regardless of the number of marbles left of each type.


For example if we have 2 red and 9 black, the chance to extract "a red marble" is `2k/(2k+9)` and the chance of extracting "a black marble" is `9/(2k+9)` --- but of course for a specific marble `i` this is `k/(2k+9)` if red and `1/(2k+9)` if black. So it could be that selecting "a black marble" is more likely, but if we look at them singularly, red are always more likely to be selected.


If I extract all the marbles, what is the most likely sequence I get, and how likely is it?


My intuition is that the most likely sequence is any of the form `red,red,...,red,black,black,...,black` (all reds in any order, then all blacks in any order), and these are all equally likely.
This is what we get by "greedily" choosing the most likely event at each step, i.e., a red marble if present, but I am not sure as to how it can be proven. Pointers to relevant theory are also appreciated.



What is the most likely sequence?
This is to note that I answered the question in the comments.


