We were hoping against hope that the judges would not notice our mistakes.
a feeling that something good will probably happen
thinking that this would happen (It may or may not have happened.)
stopped believing that this good thing would happen
wanting something to happen (and thinking it quite possible)
showing concern that what you say should not offend or disturb the other person: a way of being polite
wishing for something to happen, although this is very unlikely
Explanation
Option (F) is correct because the phrase 'hoping against hope' specifically means wishing for something to happen, although this is very unlikely. In the given sentence, the speakers are wishing that the judges will not notice their mistakes, which is a highly improbable outcome, perfectly aligning with the meaning of the idiom.