Prepostion "on"

[ian2]

I have been paying attention to the preposition "on" in sentences like:

Telephone went dead on me.
He slammed the door on me.

The action performed by the verb (went dead and slammed) is always negative, producing a negative effect on the person after the preposition.

But what does the following mean?

They took a chance on each other.

Thanks.

[Tdol]

They entered into some form of relationship though they were not sure it would work out.

[ian2]

Originally Posted by Tdol

They entered into some form of relationship though they were not sure it would work out.

But is there still some sort of negative sense in the preposition ON, as in "slam to door on me" in which ON carries a sense of unfriendliness ? In this case, negative in terms of lack of sincerity about the relationship?