Random equation-does it make sense?

Question

What is the probability that the equation $$x^2+2bx+c=0$$ has real roots? Answer is exactly $1$. (or $100$%)

For example: if $b=1$ and $c=2$ roots are complex.

Does it make sense?

If $P(A)=0$, then $A$ is an event which cannot happen, and If $P(A)=1$ then $A$ is an event which $\textbf{inevitably }$ occur. (Is this definition correct?)

what does it mean that probability equals $1$ (or $100$%)?

Answer 1

It doesn't really make sense, as pointed out by @drhab. But you could interpret the problem as follows:

let $p(R)$ be the probability that $x^2+2bx+c$ has real roots when $b$ and $c$ are chosen randomly and independently from a uniform distribution on $[-R,R]$; then find $\lim_{R\to\infty}p(R)$.

In this case we have $$p(R)=P(c\le b^2)=1-\frac1{4R^2}\int_{-\sqrt R}^{\sqrt R} R-b^2\,db =1-\frac1{3\sqrt R}$$ provided $R\ge1$, and $$\lim_{R\to\infty}p(R)=1\ .$$

Answer 2

Probability 1 doesn't been that it is inevitably. For example, the probability of throwing a dart not on a diagonal of a specific square is 1. However, it is not impossible that the dart lands exactly on the diagonal.