I raise this question following the reading of *Fifty challenging problems in probability with solutions*. One of the problem consists in computing the probability that the quadratic equation $x^2 + 2b x+c=0$ has complex roots. The "natural way" of doing it, is to suppose that the point $(b,c)$ is randomly chosen over a large square centered at the origin, with size $2B$. By following this path, the probability is equal to $0$ when $B$ approaches $+\infty$.


However, if you compute the probability over the rectangle $[-\sqrt{B},\sqrt{B}] \times [-B,+B]$, it is equal to $\frac{1}{3}$ and therefore is also the limit when $B$ approaches to $+\infty$.


I agree that the first way is "more natural". My question is: how to give more mathematical background to the "natural way"?


Thanks for your support on this "not so mathematical question".



Probability over a plane
well, to find a "natural way" to distribute the coefficients $b,c$ in the plane, you could treat this problem as the special case $n=2$ of a classic problem in random-matrix theory: take an $n\times n$ real matrix $M$ with independent identical normal distributions of the matrix elements $M\_{nm}$; what is the probability that all $n$ eigenvalues are real?


for $n=2$ we would then identify ${\rm tr}\,M=2b$ and ${\rm det}\,M=c$.


this probability is known exactly [1] for any $n$, it equals $2^{-n(n-1)/4}$; so for $n=2$ the probability of real roots is $1/\sqrt{2}$.


[1] [The circular law and the probability that a random matrix has k real eigenvalues](http://math.mit.edu/~edelman/homepage/papers/circular.pdf), A. Edelman (1993). (Published in Journal of Multivariate Analysis 60 (1997) 203-232.)


