hoffen
to hope
To have the wish or confident expectation that something positive will happen. Often used with 'auf' (for) or a 'dass' (that) clause.
Ich hoffe, dass du deine Prüfung bestanden hast.
I hope that you passed your exam.

to hope
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to hope
To have the wish or confident expectation that something positive will happen. Often used with 'auf' (for) or a 'dass' (that) clause.
Ich hoffe, dass du deine Prüfung bestanden hast.
I hope that you passed your exam.

The verb 'hoffen' (to hope) is usually constructed in two ways: with the preposition 'auf' followed by the accusative when you hope for a thing, or with a 'dass' (that) clause when you hope for an action or event.
Wir hoffen auf besseres Wetter für das Wochenende.
We are hoping for better weather for the weekend.
The verb 'hoffen' (to hope) belongs to the word family around 'die Hoffnung' (hope). If you know the noun, you can easily work out the meaning of the verb.
Solange es Leben gibt, gibt es auch Hoffnung.
As long as there is life, there is also hope.
When you hope for something, typical nouns that follow are: 'eine Besserung' (an improvement), 'ein Wunder' (a miracle), 'gute Nachrichten' (good news), or 'Unterstützung' (support).
Die Ärzte hoffen auf eine schnelle Besserung seines Zustands.
The doctors are hoping for a quick improvement in his condition.
In the general meaning 'to be confident', 'hoffen' (to hope) can also stand without an object or subordinate clause. It then describes a generally positive attitude.
Die Lage ist ernst, aber wir hoffen weiter.
The situation is serious, but we keep hoping.
The phrase 'es bleibt zu hoffen' (it remains to be hoped) is a fixed expression with an impersonal 'es' (it). It expresses a wish in an uncertain situation and is often followed by a 'dass' (that) clause.
Es bleibt zu hoffen, dass die Verhandlungen erfolgreich sein werden.
It remains to be hoped that the negotiations will be successful.
to expect
'hoffen' and 'erwarten' can both point to something in the future. The difference lies in your attitude toward it. You use 'hoffen' when you wish for something positive but do not know for sure whether it will happen. 'erwarten' more often means that you expect or assume that something will happen. This can be positive, neutral, or even negative. Learners often confuse the words because both have to do with the future. As a guide: for a wish, you are more likely to say 'hoffen'; for an assumption, 'erwarten' is more likely.

Die Ärztin hofft, dass die Behandlung schnell hilft.
The doctor hopes that the treatment helps quickly.
Am Abend erwartet der Mitarbeiter eine Antwort von seinem Chef.
In the evening, the employee expects an answer from his boss.