<s>[INST] Find the probability for having real roots with the polynomial $x^2+ax+b$ Choose two real numbers $a$ and $b$ satisfying $1<a<3$ and $-1<b<1$ randomly.
Find the probability that the polynomial $x^2+ax+b$ has two real roots.
NB: uniform distribution
 [/INST] Let $\left[x^{2}>4y\right]$ denote the function with arguments $x,y$
that takes value $1$ if $x^{2}>4y$ and takes value $0$ otherwise.
Then:
$$\begin{aligned}P\left(a^{2}>4b\right) & =\mathbb{E}\left[a^{2}>4b\right]\\
 & =\frac{1}{4}\int_{1}^{3}\int_{-1}^{1}\left[x^{2}>4y\right]dydx\\
 & =\frac{1}{4}\int_{1}^{3}\int_{-1}^{\min\left(\frac{1}{4}x^{2},1\right)}dydx\\
 & =\frac{1}{4}\int_{1}^{3}\min\left(\frac{1}{4}x^{2},1\right)+1dx\\
 & =\frac{1}{4}\int_{1}^{2}\left(\frac{1}{4}x^{2}+1\right)dx+\frac{1}{4}\int_{2}^{3}2dx\\
 & =\frac{1}{4}\left[\frac{1}{12}x^{3}+x\right]_{1}^{2}+\frac{1}{2}\\
 & =\frac{19}{48}+\frac{24}{48}=\frac{43}{48}
\end{aligned}
$$
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