First of all compare these two English sentences.

He should have arrived by now.
You should quit smoking.

In the first sentence, you are drawing a tentative conclusion; in the second, you are giving advice or a recommendation to someone. In Japanese this distinction is important as it determines the appropriate word we need to use to express “should” in English.

はず indicates a conclusion based on your current knowledge of the situation or a result that you believe is natural consequence of events.

【Verb】はず
彼は既に到着したはずです。
He should have arrived already.
赤信号を無視するのが違反だと知っているはずだよ。
You ought to know that running a red light is illegal.

You are are presuming that such knowledge is common sense.

彼女はさっき帰ってきたので、部屋にいるはずです。
She just came back a moment ago so she should be in her room.
締切が先週の火曜日だったので、xxxxxx。
xxxxxxx.

To modify nouns the possessive is required.

【Verb】はずの【Noun】
聞かなくても知っているはずのことだよ。
You should know that without having to ask.

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