Same word, two jobs. 應該2 lays a duty — this is the right thing to do (作業應該準時交). 應該1 reads off evidence to infer — this is probably so (他應該到了, he should have arrived by now). 應該2 faces what ought to be done; 應該1 faces what is likely the case.
應該 marks the proper, expected course — it ought to be done, with room to fall short (你應該道歉). 要 lays a flat requirement — it has to happen (你要道歉). 應該 weighs what is right; 要 imposes what is needed.
應該 says an action is owed by the subject (他應該來, he ought to come). 會 forecasts that an event will line up (他會來, he is likely to come). 應該 reads the duty; 會 reads the odds.
negating with 沒: 你沒應該去 → 你不應該去 (ought not to go)
putting a noun after 應該: 你應該功課 → 你應該做功課（應該 + verb for the duty）
using 應該 for a hard requirement: 坐車應該繫安全帶 (softens a rule that is mandatory) → 坐車要繫安全帶 (要 for the binding requirement)
English 'should' covers both 'it is right to' and 'it is probably so,' so learners reach for 應該 in both and blur the obligation sense (應該2) into the conjecture sense (應該1); the slot and context, not the word, separate them.