Finding the probability imaginary roots of a quadratic equation.  An urn contains 3 balls numbered 1, 2 and 3. The co-efficient of equation $px^2+qx+c = 0$
is determined by drawing the numbered balls with replacement. What is the probability that the equation will have imaginary roots?
$\textbf{Nature of roots:}$
Consider a quadratic equation: $ax^2+bx+c=0$
Compute $D=b^2-4ac<0$
$\text{Roots} = \begin{cases} D<0 & \text{imaginary roots}\\ D\geq 0 & \text{real roots}\\ \end{cases}$
I need help in solving this problem.  
I get some idea now. There are 3 numbers to be drawn with a replacement for 3 times, leaving us with $\textbf{Total } 27$ ways.
Assuming that the numbers drawn will replace $p,q,c$ in the equation:
Combination $p=1,q=3,c=2$,applined in equation $D=3^2-4.1.2<0$ will be the only combination that produce real roots.
All possible combinations of the numbers drawn will be $3! => 6$ and out of this $2$ combinations {($p=1,q=3,c=2$),($p=2,q=3,c=1$)} will produce real roots. Remaining combinations will produce imaginary roots.
So the total probability will be $=\frac{C(6,4)}{27} => \frac{15}{27}$
Is this a correct solution?
 A: You need to do some counting. Some hints for how to get started:
To start, there are $3 \cdot 3 \cdot 3 = 27$ different possibilities for the collection $(a, b, c)$, because each of those those variables can be any of three options (and they are independent of the other variables). So, in the worst case, you need to consider each of these cases separately. But, you might notice that you can make your own life easier by grouping some of the outcomes. For instance:

*

*What happens when $b = 1$? Note that this represents $9$ of the $27$ cases on its own.

*If $b = 2$, what do $a, c$ need to be in order to obtain nonnegative discriminants? This is another $9$ of the $27$ cases.

*The $b = 3$ case is the most complicated of the group, but it's not terrible; you might find it useful to write down the $9$ possibilities in this subcase and directly examine the discriminants.

Since this problem is relatively small, I'd advise just writing down cases explicitly until you're comfortable that you see the pattern(s).
