What is the probability of $\lfloor a+b \rfloor < \lfloor a \rfloor + \lfloor b \rfloor$

Question

Given $a,b \in \mathbb{R}, a\ge0, b\ge0$, what is the probability of:

$$ \lfloor a+b \rfloor < \lfloor a \rfloor + \lfloor b \rfloor $$

Answer 1

Let $\{x\}=x-\lfloor x\rfloor\geq 0$ then $$\lfloor a+b \rfloor=\lfloor a \rfloor + \lfloor b \rfloor+ \underbrace{\lfloor \{a\}+\{b\}\rfloor}_{\geq 0}\geq \lfloor a \rfloor + \lfloor b \rfloor.$$ Hence the given inequality is NEVER satisfied!