Circle and lines probability question?

Question

Given a circle, and two random lines that cut through it, what is the probability that the two lines intersect?

Answer 1

I believe your question was intersect inside the circle.
In that case answer is 1/3.
Consider 4 points of intersection of lines with circle. Every quadruplet is equally probable, but for every quadruplet when we try to split it into pairs of dots from 3 possible splittings only one correspond to intersecting lines. So total answer would be 1/3 too.

Answer 2

As mentioned above, I will assume you meant, what is the probability that the two lines intersect within the circle. I also mention that it depends how you pick the lines -- there are several different ways to pick a random line that intersects a circle.

I think the most straightforward way to pick a random line intersecting a circle is to pick two points A and B at random on the circumference, and draw the line through them. If this is the case, then the probability that AB and CD intersect is 1/3. To see why notice that AB and CD intersect if and only if C lies on one side and D on the other. Since A, B, C, D are chosen at random, their order around the circle is random and all orders are equally likely. Starting at A and moving clockwise, there are 3! = 6 possible orders of B, C, and D. 2 of those orders have B as the second point, and these are the cases when AB and CD intersect within the circle. So the probability that AB and CD intersect within the circle is 2/6 = 1/3.

Answer 3

If factors A and B are next to a minimum of one yet another the line segments do no longer go, yet whilst they have C between them on one area and D between them on the different the line segments do go. placed the 4 factors on the circle. There are 2 strategies A and B may well be adjoining and purely one way they do no longer seem to be. as a result the prospect that the line segments AB and CD go one yet another is a million/(a million + 2) = a million/3.

Answer 4

Consider the unit circle on the Cartesian plane with the center of the circle at (0).

The four axes cut the circle and also two lines most certainly intersect each other one time. Lines can only intersect at one point only!

Answer 5

you typed "2 random lines that cut THROUGH it". okay, roger that.
but you also typed "2 lines INTERSECT". this, i dont get.

if you meant "intersecting within the circle", i can give you an answer that's completely different with "intersecting either inside or outside the circle". but i'll give you both anyway.

a line cutting THROUGH a circle = connecting 2 points on the circle's perimeter(?)
2 lines cutting through = connecting 3 or 4 points on the circle's perimeter. if the 2 lines share 1 of their 2 end-points, you'll get a V-like shape in the circle. and 4 points; clearly both share not any end-points of theirs.

case 1
intersecting within (inside) the circle

if you have 4 points, say A, B, C and D consecutively, you can connect all of them to each other by 6 lines (chords) of 4 sides/outlines and 2 diagonals. of the 6 chords, you can pair them of in 6C2=15 ways.

of these 15, 1 pair is obviously "intersecting inside the circle". that's the diagonal-diagonal pair.
pair = AC - BD

of these 15, 2 pairs dont touch each other at all.
pairs =AB - CD and BC - AD

all the other (15-1-2) = 12 pairs touch each other at their shared end-points, ON the circle (and not within). the probability of "intersecting within" depends on what you define as those. if touching/sharing end-point considered as intersecting then your probability,

P(intersect within)
= (15 - 2) / 15
= 13/15

if sharing end-point's NOT intersecting,
P(intersect within)
= 1/15

case 2
intersecting lines are elongated or produced(?) outside the circle

the only way we won't get intersecting lines outside the circle is to have 2 pairs of parallel chords inside it, that is having outlining chords of rectangular shapes, including square shape (but there's yet the rectangle's inside and vertices' intersection). but for a set of 4 random points to construct a rectangular shape, that is highly unlikely. you'd need a pair of distinct diameters' end-points to get that particular shape. in short, rectangular shapes aren't that randomly acquired, so the chances are slim to none.
2 pairs of parallel lines : 13 pair of intersecting lines

trapezoidal shapes, not like rectangular shapes, will give you an intersection outside and another 1 inside.
1 pair of parallel lines : 14 pairs of intersecting lines

for other 4-sided shapes, there's 3 intersection for every shape.

but getting rectangulars/trapezoids are not common, quite a rare find actually, and even then they still have 1 inside intersection + 12 vertices intersections.

so, for
P(overall intersection)
= 1

Answer 6

Nearly 100%, seeing as them being parallel is the only instance in which they wouldn't intersect. It's 99.999999999999999999% probable that they will intersect.