I have this problem that I've struggled with for a while. If you place $4$ points randomly into a unit square (uniform distribution in both $x$ and $y$), with what probability will this shape be convex if the $4$ points are connected in some order? Equivalently, with what probability will there be a point inside the triangle with the largest area with vertices at the other $3$ points.

In particular I am interested in the answer for when this area of support is $\mathbb{R}^2$ and is uniform.

I ran a simulation and found that on a unit square the answer is about $71\%$ concave. On a unit circle picking polar co-ordinates r and theta from uniform random distributions results in a a probability of concavity of $68\%$. When the distribution for r is altered so that each point in the circle is equally likely then this falls to $51\%$.

Any advice or links for a possible answer or whether this is even possible would be appreciated.

EDIT: It turns out this problem is the same as Sylvester's 4 point problem. Alas I am 150 years too late. Thanks to all who helped. Only one person gave an answer, not quite correct but I award the bounty to them anyway for their efforts.