AMC 12 2004a Problem 20

Select numbers $a$ and $b$ between $0$ and $1$ independently and at random, and let $c$ be their sum. Let $A, B$ and $C$ be the results when $a, b$ and $c$, respectively, are rounded to the nearest integer. What is the probability that $A + B = C$?

$\text {(A) } \frac14 \text {(B) } \frac13 \text {(C) } \frac12 \text {(D) } \frac23 \text {(E) }\frac34$