YOU SAID:
They accidentally fell in.
INTO JAPANESE
彼らは誤って落ちた。
BACK INTO ENGLISH
They accidentally fell.
INTO JAPANESE
彼らは偶然落ちた。
BACK INTO ENGLISH
They fell by accident.
INTO JAPANESE
彼らは偶然に倒れた。
BACK INTO ENGLISH
They fell by accident.
That didn't even make that much sense in English.