Let $A,B,C$ be independent random variables uniformly distributed over $[0,5], [0,1] \& [0,2]$. Find probability that roots of $Ax^2+Bx+C=0$ are real.

Question

Let $A,B$ and $C$ be independent random variables, uniformly distributed over $[0,5], [0,1]$ and $[0,2]$ respectively. What is the probability that both roots of the equation $Ax^2+Bx+C=0$ are real?

I know I have to do triple integral. Here are steps, as suggested in comment.

1. Write an integral of uniform probability density over entire space of possible combinations. In this case, the "box" is rectangular and so integral bounds are trivial.
2. Find a defining equation for your condition to hold (discriminant, in our case)
3. Change integral boundaries so that you only integrate over portion where the condition is true

I am struggling in Step-3

Answer 1

You start with this: given a quadratic equation, how can you tell whether both roots are real? There is (a part of) a very well-known formula you can use.

Once you have that, you see that this condition will describe part of the box defined by the domains of $A,B,C$. Find the volume of that part, and you're very nearly done.

Answer 2

Following Polya's advice let's break this down into steps and then solve each step:

• How do we know if the roots of $ax^2 + bx + c=0$ are real ? The roots are real if the discriminant $b^2-4ac$ is greater than or equal to $0$.
• We are told $a,b,c$ are uniformly distributed within the cuboid region $0 \le a \le 5; 0\le b \le 1; 0 \le c \le 2$. So we can re-state the problem as "in what proportion of the cuboid is $b^2-4ac \ge 0$ ?".
• We know that the volume of the cuboid is $10$. So we need to find the volume of the cuboid in which $b^2-4ac \ge0$ and then divide this by $10$.
• Let's consider a simpler two-dimensional problem. Suppose $c$ is fixed at some value $C$ (so we are taking a slice of the cuboid). In what area within the rectangle $0 \le a \le 5; 0\le b \le 1$ is $b^2-4aC \ge 0$ ?
• We can re-state this again. For a fixed value $C$, in what area within the rectangle $0 \le a \le 5; 0\le b \le 1$ is $\displaystyle a \le \frac {b^2} {4C}$ ?

• Sketch a diagram. From the sketch it becomes clear that we need to find the area $A$ between the curve $\displaystyle a = \max (\frac {b^2} {4C}, 5)$ and the line $a=0$ that is also between the lines $b=0$ and $b=1$. We can do find by integration:

  $\displaystyle A(C) = \int_0^1 \max (\frac {b^2} {4C}, 5) db$

• This gives us the area $A(C)$ for a slice $c=C$ of the cuboid. To find the volume across the whole cuboid we have to integrate $A(c)$ between $c=0$ and $c=2$. Then divide by 10 to find the answer to the original problem.