Consider the quadratic function $$f(x) = ax^2 + bx + a$$ having two irrational roots, with a and b being two positive integers, such that $$a, b \leq 9$$.
If all such permissible pairs (a, b) are equally likely, what is the probability that a + b is greater than 9?

For the roots to be irrational, the value of D > 0 and D must not be a perfect square.

The value of D for the equation given is $$=\ b^2\ -\ 4\times\ a\ \times\ a\ =\ b^2\ -\ 4a^2$$

The pairs (a. b) that satisfy the above conditions are,

b =  1 no value of a exists

b = 2 no value of a exists

b = 3, a = 1

b = 4, a = 1

b = 5, a = 1

b = 6, a = 1, 2

b = 7, a = 1, 2, 3

b = 8, a = 1, 2, 3

b = 9, a = 1, 2, 3, 4

There are a total of 15 pairs of (a, b) that satisfy the above condition and out of them, the value of a + b is greater than 9 for 7 of them.

So, the probability that a + b is greater than 9 is given by $$\dfrac{7}{15}$$.

Hence, the correct answer is option D.