Question

If $p$ and $q$ are chosen randomly from the set $\{1,2,3,4,5,6,7,8,9$ and $10\}$ with replacement, determine the probability that the roots of the equation $x^2+px+q=0$ are real.

Answer 1

The required probability $=1-$ (probability of the event that the roots of $x^2+px+q=0$ are non-real).
The roots of $x^2+px+q=0$ will be non-real if and only if $p^2-4q<0$, i.e ., if $p^2<4q$
The possible values of $p$ and $q$ can be possible according to the following table

Therefore, the number of possible pairs $=38$
Also, the total number of possible pairs is $10\times 10=100$
$\therefore$ The required probability = $1-\frac{38}{100}=1-0.38=0.62$