I have 3 cups. Cup 1 can hold 1 ml of water. Cup 2 can hold 5 ml of water. Cup 3 can hold 10 ml of water. I only have 12 ml of water.
If I distribute water to each cup randomly what is the average amount of water in each cup once I’ve distributed all the water?
I tried to set this up as a set of algebraic equations like:
C1 + C2 + C3 = 12
0 ≤ C1 ≤ 1
0 ≤ C2 ≤ 5
0 ≤ C3 ≤ 10
Then solving numerically. We assume the water is infinitely divisible and because of this there seems to be an infinite set of solutions to these equations. When I change the starting point of the numerical solver I often get results around:
C1 = 0.833
C2 = 3.833
C3 = 7.333
Is there a way to calculate the exact statistical averages?
The equation and three inequalities together define a rectangle with vertices $(c_1,c_2,c_3) = (1,1,10)$ and $(0,2,10)$ and $(0,5,7)$ and $(1,4,7)$, so perhaps you mean the distribution should be uniform on that rectangle; in that case the average values would be $(0.5,3,8.5)$ respectively (the center of the rectangle). The fact that you're getting values close to $(\frac56,3\frac56,7\frac13)$ suggests, however, that you're using some different distribution or sampling method. Without defining exactly what "randomly" means, it's impossible to answer this question.