If I generate a random symmetric matrix, what's the chance it is positive definite? I got a strange question when I was experimenting some convex optimizations. The question is: 
Suppose I randomly (say standard normal distribution) generate a $N \times N$ symmetric matrix, (for example, I generate upper triangular matrix, and fill the bottom half to make sure it is symmetric), what is the chance it is a positive definite matrix? Is there anyway to calculate the probability?
 A: If your matrices are drawn from standard-normal iid entries, the probability of being positive-definite is approximately $p_N\approx 3^{-N^2/4}$, so for example if $N=5$, the chance is 1/1000, and goes down quite fast after that. You can find an extended discussion of this question here.
You can somewhat intuit this answer by accepting that the eigenvalue distribution of your matrix will be approximately Wigner semicircle, which is symmetric about zero. If the eigenvalues were all independent, you'd have a $(1/2)^N$ chance of positive-definiteness by this logic. In reality you get $N^2$ behavior, both due to correlations between eigenvalues and the laws governing large deviations of eigenvalues, specifically the smallest and largest. Specifically, random eigenvalues are very much akin to charged particles, and do not like to be close to each other, hence they repel each-other (strangely enough with the same potential field as charged particles, $\propto 1/r$, where $r$ is the distance between adjacent eigenvalues). Asking them to all be positive would therefore be a very tall request.
Also, because of universality laws in random matrix theory, I strongly suspect the above probability $p_N$ will likely be the same for essentially any "reasonable" random matrix, with iid entries that have finite mean and standard deviation. 
