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.

He should have arrived already.
You ought to know that running a red light is illegal.

You are are presuming that such knowledge is common sense.

She just came back a moment ago so she should be in her room.

To modify nouns the possessive is required.

You should know that without having to ask.

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