Question

What is the probability that all the roots of x2 + bx + c = 0 are real? [0,1]?

Answer 1

`x^2+bx+c` will have real roots when the discriminant of the quadratic,
`\Delta=b^2-4c`, is non-negative, i.e.

`b^2-4c\ge0\implies b^2\ge4c`

Your question about probability is currently impossible to answer without knowing exactly what the experiment is. Are you picking
`b,c` at random from some interval? Is the choice of either distributed a certain way?

I'll assume the inclusion of "[0,1]" in your question is a suggestion that both
`b,c` are chosen indepently of one another from [0, 1]. Let
`B,C` denote the random variables that take on the values of
`b,c`, respectively. I'll assume
`B,C` are identical and follow the standard uniform distribution, i.e. they each have the same PDF and CDF as below:

`f_X(x)=\begin{cases}1&\text{for }0<x<1\\0&\text{otherwise}\end{cases}`

`F_X(x)=\begin{cases}0&\text{for }x<0\\x&\text{for }0\le x<1\\1&\text{for }x\ge1\end{cases}`

where
`X` is either of
`B,C`.

Then the question is to find
`P(B^2\ge4C)`. We have

`P(B^2\ge4C)=P\left(C\le\frac{B^2}4\right)`

and we can condition the random variable
`C` on the event of
`B=b` by supposing

`P\left(C\le\frac{B^2}4\right)=P\left(\left(C\le\frac{B^2}4\right)\land(B=b)\right)=P\left(C\le\frac{B^2}4\mid B=b\right)\cdot P(B=b)`

then integrate over all possible values of
`b`.

`=\displaystyle\int_(-\infty)^\infty P\left(C\le\frac{b^2}4\right)f_B(b)\,\mathrm db`

`=\displaystyle\int_(-\infty)^\infty F_C\left(\frac{b^2}4\right)f_B(b)\,\mathrm db`

`=\displaystyle\int_0^1\frac{b^2}4\,\mathrm db=\frac1{12}`