From Longman Dictionary of Contemporary Englisha faint hope/possibility/chance etca faint hope/possibility/chance etcSMALLa very small or slight hope etc a faint hope that they might be alive → faint
Examples from the Corpus
a faint hope/possibility/chance etc• I thought about letting it ring, but there was a faint hope that it might be Sally.• There remained a faint possibility that Newley would try to identify the person who collected the money.• If it can startle the predator in some way, there is a faint chance that the enemy may panic and flee.• That uncertainty urges us to look beyond the present, with a faint hope to control our future. 