Probability and N-Dimensions

Hi there, i was looking to get some insight on the following problem.

Question — What is the probability of n randomly drawn lines intersecting at a point in 2-D space. ? Stated in a different way, what is the probability that a randomly chosen system of n equation in 2 variables yields a result.

Follow up — does this probability equals the probability of n random planes intersecting at a point in 3-D space ? more generally, does this equal to the probability of n random N-dimension hyperplanes intersecting at a point in N-D space ?

I did googled things, but i found this question stacked with some constraint on space, for example, in THIS the space is constrained to a circle and in THIS the spaces is a 3-D grid. these made me thing whether there exist something for general case or not. Please Help. Feel free to ask for any further clarification about the setting of the problem ?