Let me give a reasonable model for the question in the title. In ${\rm Sym}_n({\mathbb R})$, the positive definite matrices form a convex cone $S_n^+$. The probability I have in mind is the ratio $p_n=\theta_n/\omega_n$, where $\theta_n$ is the solid angle of $\Lambda_n$, and $\omega_n$ is the solid angle of the whole space ${\rm Sym}_n$ (the area of the unit sphere of dimension $N-1$ where $N=\frac{n(n+1)}2$). These definitions are relative to the Euclidian norm $\|M\|=\sqrt{{\rm Tr}(M^2)}$ ; this is the most natural among Euclidian norms, because it is invariant under unitary conjugation.
Because $S_2^+$ is a circular cone, I could compute $p_2=\frac{2-\sqrt2}4\sim0.146$ . Is there a known close formula for $p_n$? If not, is there a known asymptotics?
More generally, we may define open convex cones
$$\Lambda_n^0\subset\Lambda_n^1\subset\cdots\subset\Lambda_n^{n-1}$$
in the following way: $M\mapsto\det M$ is a homogeneous polynomial, hyperbolic in the direction of the identity matrix $I_n$. Thus its successive derivatives in this direction are hyperbolic too. The $k$th derivative defines a "future cone" $\Lambda_n^k$, these cones being nested. For instance, $\Lambda_n^0=S_n^+$. It turns out that this derivative is, up to a constant, $\sigma_{n-k}(\vec\lambda)$, where $\sigma_j$ is the $j$th elementary symmetric polynomial and $\vec\lambda$ the spectrum of $M$. Therefore $\Lambda_n^k$ is defined by the inequalities
$$\sigma_1(\vec\lambda)\ge0,\ldots,\sigma_{n-k}(\vec\lambda)\ge0.$$
For instance, $\Lambda_n^{n-1}$ is the half-space defined by ${\rm Tr}M\ge0$.
Let us define again $p_{n,k}$ the probability for $M\in{\rm Sym}_n$ to belong to $\Lambda_n^k$. Thus $p_{n,0}=p_n$ and $p_{n,n-1}=\frac12$.
What is the distribution of $(p_{n,0},\ldots,p_{n,n-1})$, asymptotically as $n\rightarrow+\infty$?
Edit. As Mikaël mentionned, it is equivalent, and easier for calculations, to consider the standard Gaussian measure (GOE) over ${\bf Sym}_n$.
Best Answer
Edit: According to Dean and Majumdar, the precise value of $c$ in my answer below is $c=\frac{\log 3}{4}$ (and $c=\frac{\log 3}{2}$ for GUE random matrices). I did not read their argument, but I have been told that it can be considered as rigourous. I heard about this result through the recent work of Gayet and Welschinger on the mean Betti number of random hypersurfaces. I am a bit surprised that this computation was not made before 2008.
Let me just expand my comment. You are talking about the uniform measure on the unit sphere of the euclidean space $Sym_n(\mathbb R)$, but for measuring subsets that are homogeneous it is equivalent to talk about the standard gaussian measure on $Sym_n(\mathbb R)$. This measure is called in random matrix theory the Gaussian Orthogonal Ensemble (GOE). In particular $p_n$ is the probability that a matrix in the GOE is positive definite. Since there are explicit formulas for the probability distribution of the eigenvalues of a GOE matrix (this is probably what Robert Bryant is proving), there migth be explicit formulas for $p_n$.
Anyway, the asymptotics are known from general large deviation results for random matrices (due to Ben Arous and Guionnet, PTRF 1997)~: $p_n$ goes to zero as $e^{-c n^2}$ for some constant $c>0$. The constant is equal to the infimum, over all probability measures $\mu$ on $\mathbb R^+$, of the quantity $$ \frac{1}{2} (\int x^2 d\mu(x) - \Sigma(\mu)) - \frac 3 8 - \frac 1 4 \log 2$$ where $\Sigma(\mu)$ is Voiculescu's free entropy $\iint \log|x-y| d\mu(x) d\mu(y)$. You can probably explicitely compute $c$. It is even possible that this was known before Ben Arous and Guionnet's work, since their results are much more general.
For your second question, I am pretty sure that the limiting graph of $t \in [0,1( \mapsto p_{n,E(tn)}$ is $0$ ($E(x)$ is the integer part of $x$). But this is probably not what you really want to ask.