Ok, let’s do the probability math properly. Others have mentioned how it’s a matter of probability how long until the last molecule of soup is taken out.

Suppositions:

There are N molecules in every ml of soup and every ml of water.

All soup molecules are the same.

Every spoonful takes out exactly 25N molecules out of the bowl selected at random, and they are immediately replaced by 25N molecules of water.

At the start, there are 500N molecules of soup in the bowl.

The question is:

How many spoonfuls is it expected to take until all soup molecules are removed?

For every spoonful, each molecule of soup in the bowl has a 25/500 chance of being removed from the bowl.

For ease of calculation, I will assume that each molecule being removed is independent of all others. This is technically wrong, because this implies that there is a very very tiny chance that all soup molecules are replaced in the very first spoonful. However, for the large number of molecules we are going to be working with, this shouldn’t affect the final result in any meaningful way.

Number all soup molecules in the bowl: 1, 2, …, 500N.

Define X_i to be the number of iterations it took until molecule i was removed. All X_i are I.I.D.:

P(X_i = 1) = 25/500 P(X_i = 2) = (475/500) * 25/500 P(X_i = 3) = (475/500)² * 25/500 … P(X_i = n) = (475/500)^(n-1) * 25/500

This is a geometric distribution with p = 25/500.

Now what we’re interested in if the maximum value between all X_i

That is: max_i { X_i }

Specifically we want the “Expected Value” (basically the average) of it: E[ max_i { X_i } ]

This is exactly the question asked here: math.stackexchange.com/q/26167

According to the answer there, there is no closed-form exact answer but a very good approximation for the solution is:

1/2 + (1/λ) H_N

Where λ = -log(1-p) and H_n is the nth harmonic number.

Now it’s just a matter of plugging in the numbers.

According to Wolfram Alpha, there are N = 3.333*10^22 molecules in 1mL of water.

Again using Wolfram Alpha, the Nth harmonic number is H_N = 52.438

With the formula given we get λ = -log(475/500) = 0.051293

Plugging it all in we get the expected number of spoonfuls:

1022.8