Consider a square space.
Randomly select 4 points.
Randomly connect two sets of two to each other with line segments.
What is the chance of the line segments intersecting?
(I will maybe try and solve this with a little Monte-Carlo simulation but would be very interested in an analytic solution.)
Let's be perfectly clear about the assumptions. I assume that "randomly selecting 4 points" A,B,C,D in the unit square means selecting uniformly and independently at random their coordinates $x_A,y_A,x_B...$ in the $[0,1]$ range; and that one of the 6 possible pairings between points to form the segments is also chosen uniformly and independently at random -- so the problem is equivalent to asking what is the probability that, e.g. AC and BD intersect.
It's easy to see that the solution is $\frac{25}{108}$, or about $23\%$, if one is willing to take as a given the solution for Sylvester's four-point problem for the square -- the probability that the convex hull of four points chosen uniformly and independently at random in a square is a quadrilateral, which is $\frac{25}{36}$. It is immediate to see that two segments chosen by a random pairing of four points intersect if, and only if, the convex hull of the four points is indeed a quadrilateral, and the two segments are its diagonals. If the former condition is satisfied, the latter happens with probability $\frac{1}{3}$ (the probability that a given point is paired with "middle one" of the remaining three), yielding a solution of $\frac{25}{36}\cdot\frac{1}{3}=\frac{25}{108}$ for the original problem.
Sylvester's four-point problems allows one to obtain the answer in exactly the same way even if the points are chosen uniformly at random in another convex region, simply by multiplication by $\frac{1}{3}$. The general formula for Sylvester's four-point problem in a convex region $R$, as given in the above link, equals $1-4\frac{\bar{A_R}}{A(R)}$ where $A(R)$ is the area of the region (in the case of the unit square, $1$) and $\bar{A}_R$ equals the expected area of a triangle obtained from $3$ points chosen uniformly at random within it. The formula is easily obtained noting that probability that one given point lies within the triangle formed by the other $3$ is $\frac{\bar{A_R}}{A(R)}$, and since the events of this happening for different points are mutually exclusive, the probability that no point lies within the triangle formed by the other three is indeed $1-4\frac{\bar{A}_R}{A(R)}$.
Computing $\frac{\bar{A_R}}{A(R)}$ is non-trivial, in general. If R is a square (our case), a simple proof that $\frac{\bar{A_R}}{A(R)}=\frac{11}{144}$ can be found here. More in general if R is regular polygon of $n$ sides the solution is given by Alikosky's formula, $\frac{\bar{A_R}}{A(R)}=\frac{9\cos^2 (2\pi/n)+52\cos(2\pi/n)+44}{36n^2\sin^2(2\pi/n)}$. Note that this is a strictly decreasing function of $n$, so the intersection probability is strictly increasing with $n$, and the limit for $n\to\infty$, i.e. $\frac{35}{48\pi^2}$, yields the solution when picking points in the circle.
So, in general, the probability of intersection of $2$ segments, each made by $2$ points chosen uniformly and independently at random from a regular $n$-agon or any of its affine transformations (such as a rectangle or parallelogram) is:
$\frac{1}{3}\left(1-4\cdot\frac{9\cos^2(2\pi/n) + 52\cos(2\pi/n) + 44}{36 n^2\sin^2 (2\pi/n)}\right)$
with the limit for $n\to\infty$, i.e. $\frac{1}{3}-\frac{35}{36\pi^2}$, yielding the probability when choosing the points from a circle (or any affine transformation, such as an ellipse).
First let's select 4 points inside a square. Let $P_c$ be a probability that these 4 points form a convex figure (none of the selected points is inside the triangle formed by the other three).
If this is the case, the probability that two line segments intersect is 1/3.
Otherwise the probability is 0.
So, the probability we are looking for is:
$P=1/3 * P_c$
Now we need to find the $P_c$.
Let's calculate the probability $P_1$ that the first point is inside the triangle formed by the three others. It is $P_1 = S_{234}/S_{square}$, where $S_{234}$ is the expected area of the triangle formed by points 2, 3 and 4.
Same for $P_2$, $P_3$ and $P_4$. Obviously $P_1=P_2=P_3=P_4$.
Probability that any point is inside of triangle formed by other 3 is $P_1+P_2+P_3+P_4$ because no more than 1 of these outcomes can happened simultaneously.
$P_c = 1 - 4*P_1 = 1 - 4 * S_{234}$
And now we only need to find $S_{234}$ - expected area of a triangle formed by randomly selected 3 points inside a square.
The expected area of a triangle formed by three points randomly chosen from the unit square
They say it is $11/144$.
$P=1/3 * (1 - 4*11/144) = 25/108$
This is in a good agreement with my experiments.
If in four points if $$\color{red}{1]}$$ all are collinear, line segments will never 'intersect' $$\color{red}{2]}$$ Three are collinear, still no intersection. $$\color{red}{3]}$$ Two points are always collinear:) $$$$ $$..................................$$$$$$ 4 non collinear points will form quadrilateral.
$$\color{red}{\text{if quadrilateral is convex}}$$ Lines segments drawn must be either sides or diagonals.
Only diagonals intersect.$$$$ $$\color{red}{\text{if quadrilateral is concave}}$$ No intersection
Solve using this information
This is my approach:
Consider a line $y=mx+c$ that divides the unit square into two trapeziums. Let us call the left trapezoid $a$, and the right trapezoid $b$.
By solving the equations $(y=mx+c)=1$ and $(y=mx+c)=0$, or $mx+c=1$ and $mx+c=0$, the $x$-values of the intersections are $x = \frac{-c+1}{m}$, and $\frac{-c}{m}$. Since the $y$-values are $0$ and $1$, the intersection coordinates are $(\frac{-c}{m}, 0)$ and $(\frac{-c+1}{m}, 1)$.
Using this information, the area of trapezium $a$ is $1 * \frac{(-c/m+1)+(-c/m)}{2}$, and of $b$ is $1 - (1 * \frac{(-c/m+1)+(-c/m)}{2})$ (since $a+b=1$). Simplifying gives the area of $a$ as $-\frac{c}{m}+ \frac{1}{2}$, and the area of $b$ as $\frac{c}{m}+\frac{1}{2}$.
For the line connecting two points to not cross the green line, the points cannot be in $a$ and $b$. Since using geometric probability, the probability of this is $ab-ba$, we can subtract this condition from $1$ to get $1 - 2(-\frac{c}{m}+ \frac{1}{2})(\frac{c}{m}+\frac{1}{2})$, and after simplifying we have the probability as $1 - 2(-\frac{c^2}{m^2}+\frac{1}{4})$ or $$\frac{2c^2}{m^2}+\frac{1}{2},$$
where $c$ is the $y$-intercept of the line (when extended), and $m$ is its slope.
Note: This example doesn't work when $c=0$, because regardless of what $m$ is chosen, it would equal $0+\frac{1}{2}=\frac{1}{2}$, and when the line crosses a point not between $0$ and $1$ when $x=0$ or $x=1$.
