Ignoring the premise of probability question

My friend suggested a question in geometric probability:

What is the probability of a random line A's end point, to not meet another random line B's end point? (in a space where only these two lines exist.)

The answer I suggested: The probability of  point A falling exactly on another point B in space is low. (one chance in infinity). Therefore the probability of them not meeting is high (infinity minus one chance in infinity)

I could be wrong about the statistics, but I'll try to get to the actual logical problem:

Friends's answer was: That is a logical error based on cognitive bias. Point B has no more significance than any other point in space. Therefore the probability of the end point A, meeting exactly the endpoint B, is equal to end point A falling on any other random point in the space.

It seems to me that this is ignoring the basic premise of the question. The question wouldn't even exist if there wasn't a specific point B to talk about! that's the whole point of the question to be asked in the first place.

Is there any merit to her claim? Is there a logical fallacy in her answer and if so what is it?