6. Two real numbers $a$ and $b$ are randomly chosen from $[-1,1]$. Then the probability that the equation $x^{2}+a x+b=0$ has real roots is $\qquad$
6. $\frac{13}{24}$.

Notice,
the equation $x^{2}+a x+b=0$ has real roots
$$
\Leftrightarrow a^{2}-4 b \geqslant 0 \Leftrightarrow b \leqslant \frac{1}{4} a^{2} \text {. }
$$

Thus, in the Cartesian coordinate system $a O b$, the set of real number pairs $(a, b)$ that satisfy the condition forms the shaded region in Figure 1, with an area of
$$
S=2+2 \int_{0}^{1} \frac{1}{4} a^{2} \mathrm{~d} a=\frac{13}{6} .
$$

Therefore, the required probability is $p=\frac{S}{4}=\frac{13}{24}$.