jmc

The conditions under which $A+B=C$ are as follows.

(i) If $a+b<1/2$, then $A=B=C=0$.

(ii) If $a\ge 1/2$ and $b<1/2$, then $B=0$ and $A=C=1$.

(iii) If $a<1/2$ and $b\ge 1/2$, then $A=0$ and $B=C=1$.

(iv) If $a+b\ge 3/2$, then $A=B=1$ and $C=2$.

These conditions correspond to the shaded regions of the graph shown. The combined area of those regions is 3/4, and the area of the entire square is 1, so the requested probability is $\boxed{\frac{3}{4}}$.