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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".

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  • $\begingroup$ On line 3 you probably mean "real roots" instead of "complex roots"? $\endgroup$
    – Qfwfq
    Commented Jun 29, 2014 at 17:30
  • $\begingroup$ In fact the original question in the book is indeed "real roots". I changed to "complex roots" as it simplifies some computations. $\endgroup$
    – mathcounterexamples.net
    Commented Jun 29, 2014 at 18:01
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    $\begingroup$ Ok.. so I'm surely missing something. I would naively think that the probability that a polynomial has a complex root is always $1$ due to the fundamental theorem of algebra... $\endgroup$
    – Qfwfq
    Commented Jun 29, 2014 at 19:04
  • $\begingroup$ By complex, I mean not real, which should avoid the confusion. $\endgroup$
    – mathcounterexamples.net
    Commented Jun 29, 2014 at 20:54

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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, A. Edelman (1993). (Published in Journal of Multivariate Analysis 60 (1997) 203-232.)

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    $\begingroup$ Thanks Carlo. What I take from that discussion is that indeed the question: "probability that the quadratic equation $x^2 +2bx+c=0$ has complex roots" needs to be further detailed in order to give to it a precise meaning. $\endgroup$
    – mathcounterexamples.net
    Commented Jun 29, 2014 at 14:52

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