# Probability of getting a fill of a given size

Question is quite simple, what is the probability of getting a fill for:

(i) a limit order of size - $$s$$

(ii) that is $$\Delta$$ away from the mid-price

(iii) in the next $$t$$ minutes

given empirical data of the market orders (prices and sizes of the fills, as well as the time of the fills). I think the Poisson process is useful here?

I have found this. Although it asks this question for the longer-timeframe, I suppose my question is a generalization of that question.

I would like to know this to optimally place the limit orders with varying sizes at varying distances from the mid-price to obtain the largest expected value of the fills. It would also be useful to know the expected order size at a given distance from the mid-price.

Also, according to this, what I am attempting to do might be nonsensical? And I should just base placing of the limit orders on game theory?