2004 AMC 12A 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 \qquad \text {(B) } \frac13 \qquad \text {(C) } \frac12 \qquad \text {(D) } \frac23 \qquad \text {(E) }\frac34

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.