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?
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Use the discriminant to identify conditions for irrational roots, and carefully count the pairs that satisfy the given condition.
Step 1: Condition for irrational roots. A quadratic equation yields irrational roots if its discriminant is positive and not a perfect square. The discriminant for \( f(x) = ax^2 + bx + a \) is calculated as:\[\Delta = b^2 - 4ac = b^2 - 4a^2\]The discriminant \( \Delta \) must satisfy the conditions of being positive and not a perfect square. Step 2: Determine the count of valid \( a \) and \( b \) values. Given that \( a \) and \( b \) are positive integers not exceeding 9, there are 9 possible values for each, resulting in 9 potential values for \( a \) and 9 for \( b \). Step 3: Compute the probability. The probability is derived from the quantity of valid pairs \( (a, b) \) where the sum \( a + b \) exceeds 9. Final Answer: \[\boxed{\frac{2}{3}}\]