I think you guys actually agree more than you think --

You say "you cannot fully describe 1 student at table of three people as 1/3", which is true: What's missing is the unit (or dimension, I'm ignoring the difference here).

You can only add two things if they have the same units, as per "dimensional analysis" [1].

So this is an entirely meaningless statement:

[students]/[seats at table 1] + [students]/[seats at table 2]

But you can fix the units with some multiplication (because dimensions do form an Abelian group under multiplication):

([students]/[seats at table 1]) * ([seats at table 1]/[total seats]) + ([students]/[seats at table 2]) * ([seats at table 2]/[total seats])

Which simplifies to:

[students]/[total seats] + [students]/[total seats]

Now that's a statement with meaning!

Since I know that

[seats at table 1]/[total seats] = 1/2

[seats at table 2]/[total seats] = 1/2

I've just derived the calculation that I really wanted to do:

(1/3)(1/2) + (1/3)(1/2) = (2/6)

[1] https://en.wikipedia.org/wiki/Dimensional_analysis#Dimension...