If p and q are chosen randomly from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} with replacement, then the probability that the roots of the equation x² + px + q = 0

Roots of x² + px + q = 0 will be real if ${\mathrm{p}}^2\ge 4\mathrm{q}$.

The possible selections are as follows:

Therefore, number of favourable ways is 62 and total number of ways is 10² = 100. Hence, the required probability is

62/100 = 31/50.

The probability that the roots are imaginary is

1 - 31/50 = 19/50.

Roots are equal when (p, q) $\equiv$ (2, 1), (4, 4), (6,9). The probability that the roots are real and equal is 3/50. Hence, probability that the roots are real and distinct is 3/5.