First, since the problem is obviously rescale and rotationally invariant we can assume that the four points are chosen uniformly at random inside the unit disk $\newcommand{\bD}{\mathbb{D}}$ $\bD$. Denote by $V$ the number of points vertices of the convex hull of these four random points. Note that $V$ is a random variable and $V\in\{3, 4\}$ almost surely. $\newcommand{\bE}{\mathbb{E}}$ $\newcommand{\bP}{\mathbb{P}}$ We set
$$
p:=\bP[V=4].
$$
Note that $\bP[V=3]=1-p$ so the expectation $\bar{v}$ of $V$ is
$$
\bar{v}= 3(1-p)+4p=3+p,
$$
so
$$
p=\bar{v}-3.
$$
The computation of $\bar{v}$ is a special case of the Renyi-Sulanke problem.

Denote by $P_1,P_2,P_3,P_4$ the $4$ random points and by $p_{12}$ the probability that the points $P_3,P_4$ lie on the same side of the line determined by $P_1,P_2$. Then, as shown by Renyi and Sulanke we have
$$
\bar{v}=\binom{4}{2}p_{12}= 6p_{12}.
$$
Given two points $P,Q$ we denote by $L(P,Q)$ the line determined by $P,Q$ and by $a(P,Q)$ the area of the part of the disk that lies on the same side of $L(P,Q)$ as the center of the disk. Set
$$
\bar{a}(P,Q)=\frac{1}{\pi} a(P,Q).
$$ Then
$$
p_{12}= \frac{1}{\pi^2}\int_{\bD^2}\Big(\; \bar{a}(P,Q)^2+(1-\bar{a}(P,Q))^2\;\Big)dPdQ
$$
$$=\int_{\bD^2}\Big(\; 2\bar{a}(P,Q)^2-2\bar{a}(P,Q)+1\;\Big) dPdQ $$
$$
=1-2\int_{\bD^2} \bar{a}(P,Q)\big(\; 1-\bar{a}(P,Q)\;\big) dPdQ.
$$
The line $L(P,Q)$ has a unique equation of the form
$$
x\cos \theta+y\sin\theta =p,\;\;p\in[0,1], \;\;\theta\in[0,2\pi].
$$
We denote by $L_{\theta,p}$ the line described by such an equation. The chord cout-ut by the disk on this line has length $$\ell(p)=2\sqrt{1-p^2}.$$ For such a line we have
$$
a(P,Q)=\frac{\pi}{2}+2\int_0^p \sqrt{1-x^2} dx =\frac{\pi}{2} +p\sqrt{1-p^2} +\arcsin(p).
$$
We set
$$
\alpha(p):=\frac{1}{2}+ \frac{1}{\pi} \big(\; p\sqrt{1-p^2} +\arcsin(p)\;\big)=\bar{a}(P,Q).
$$
As explained in the equality (5) in here we have
$$
\int_{\bD^2} \bar{a}(P,Q)\big(\; 1-\bar{a}(P,Q)\;\big) dPdQ=\frac{1}{3}\int_0^{2\pi} \int_0^1\alpha(p)(1-\alpha(p)) \ell(p)^3 dp d\theta,
$$
$$
=\frac{2\pi}{3} \int_0^1 \alpha(p)(1-\alpha(p)) \ell(p)^3 dp.
$$
This integral can be computed/approximated using a computer algebra system such as MAPLE or MATHEMATICA.