# What is the likelihood that n infinite lines intersect at one point? [on hold]

1)Assuming all lines are coplanar. or 2)Assuming all lines are coplanar and nonparallel.

I would like to see an answer for both if possible. If not, the more constrained outcome for #2 will work.

Thank you!

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## put on hold as off-topic by Lord Shark the Unknown, YiFan, Cesareo, José Carlos Santos, Kemono ChenFeb 12 at 9:24

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• What do you mean by "likelihood"? Some sort of probability measure? – Lord Shark the Unknown Feb 12 at 5:36
• Correct. Perhaps another way to think of it if this is more clear: What is the probability that N infinite lines intersecting at one point is random versus intentional? Also it doesn't necessarily need to be infinite. That was purely for curiosity. It could be say 10 infinite lines intersecting at one point. What is the probability of that occurrence happening randomly versus someone intentionally plotting the intersection. – Asteroid Nap Feb 12 at 5:57
• I’m an outsider to probability theory, but I would think that to speak of probability here, you need to specify both the space you’re talking about (all lines in the plane, perhaps?) and the probability measure on it. Till you do that, nothing whatever can be said. – Lubin Feb 12 at 6:45
• I did say that the lines are coplanar. I've also approached the question from different angles though I haven't mentioned them yet. I've considered it the form of uncooked spaghetti sticks (I was hungry) that one might drop on the floor. If you lay one spaghetti stick down (and pretend it's infinite), then it seems to me you cannot determine whether one intersection point made by a 2nd piece of spaghetti is from a random drop or placed deliberately. In fact, two pieces of spaghetti being parallel by random drop is far less likely. I believe that chance is something like 1/180. – Asteroid Nap Feb 12 at 7:25
• If two infinite sticks are placed randomly, the chance they are parallel is similar to the problem of rolling doubles on a pair of dice - except instead of 1 to 6, it's 0 - 180 (as we don't care which end of the stick). The parallel cases are trivial. But as more sticks are added, the likelihood of all the sticks intersecting at one point because of chance approaches zero. Intuitively this makes sense, but I'm looking for a mathematical expression if possible. – Asteroid Nap Feb 12 at 7:35