If I have 4 different balls and 4 persons. Each person should choose only one ball. Each person will choose the ball according to an equation. That equation simply means that the person should choose the ball that he is too close to which means "if person 1 is too close to ball 1, then he will probably choose that ball and he will not consider choosing the
other balls because he is far from them"

My question is that:
I can get the probability of any event, but I know that the equation that makes the person to choose the closest ball should be incorporated in the probability of the events, so how can I engage it.
Note the equation have one random variable which determines the distance between the ball and the person.

If the placement of balls and people is random, so that the ball closest to each person is random, then it really doesn't matter - you approach this like any other problem of selection without replacement. But if the placement of balls relative to people is governed by some law as expressed by the equation, then it gets a bit complicated. Perhaps if you share with us what the equation is we can be more specific.