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.



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> 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?
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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 $p\_{n,n-1}=\frac12$. 



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> What is the distribution of $(p\_{n,0},\ldots,p\_{n,n-1})$, asymptotically as $n\rightarrow+\infty$?
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**Edit**. As Mikaël mentionned, it is equivalent, and easier for calculations, to consider the standard Gaussian measure (GOE) over ${\bf Sym}\_n$.



The probability for a symmetric matrix to be positive definite
**Edit:** According to [Dean and Majumdar](http://arxiv.org/abs/0801.1730), 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](http://arxiv.org/abs/1207.1579) on the mean Betti number of random hypersurfaces. I am a bit surprised that this computation was not made before 2008.




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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.


