3. If two real numbers $a$ and $b$ are randomly chosen from the interval $(0,1)$, find the probability that the equation $x^{2}-\sqrt{a} x+b=0$ has real roots.
(A) $\frac{1}{8}$
(B) $\frac{5}{16}$
(C) $\frac{3}{16}$
(D) $\frac{1}{4}$
(E) $\frac{1}{3}$
3. Answer: (A)
The discriminant of the equation is $a-4 b$. Thus the equation has real roots if and only if $a \geq 4 b$. The shaded part in the figure on the right are all the points with coordinates $(a, b)$ such that $0<a, b<1$ and $a \geq 4 b$. As the area of the shaded part is $\frac{1}{8}$, it follows that the required probability is $\frac{1}{8}$.