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$