The pen is a circle or a square? It looks like a circle, but if it were a square it may be easier to calculate the probabilities.

(I guess there is no closed form for the probabilities, but if the pen is a square I guess ¿¿¿P(x,y)=P(x)*P(y)??? or something like that?

Do you have some "heat maps" of frequencies after 1000 simulations?

A related problem the limit of the probabilities of finding the pen in one point. It's probably greater in the center, because when it's near the border it got sucked and teletransported to the center. It also may have strips, instead of being a function that just decrease when move from the center.