The quadratic equation

$4x^2+bx+c=0$

has equal roots if $b^2-16c=0$

$\Rightarrow$

$b^2=2^{4}c$

Now, C should be chosen from the set

$S=\{1,2,3,\dots ,100\},$ such that it is a perfect square number, $c=1,4,9,16,25,36,49,64,81,100$,49,64,81,100$

$\therefore$ number of ordered pair $(b, c)$ will be 10 . So, required probability $=\frac{10}{100\times 100}$

$=\frac{1}{1000}=0.001$