Each coefficient in the equation $ax^2 + bx + c = 0 $ is determined by throwing an ordinary six faced die. The probability that the equation will have real roots, is

$\frac{43}{216}$

Since each of the coefficient a, b and c can take the values from 1 to 6.

∴ Total numbers of equations = 6 × 6 × 6 = 216.

The roots of the equation $ax^2 + bx + c = 0 $ will be real if $b^2 - 4 ac ≥ 0 ⇒ b^2 ≥ 4 ac.$

The  favourable number of elementary events can be enumerated as follows:

                                                         Total = 43

Since $b^2≥ 4ac$ and since the maximum value of $b^2$ is 36, therefore ac = 10, 11, 12 ...etc. is not possible.

∴ Total number of favourable elementary events = 43

Hence, required probability = $\frac{43}{216}$