About Da'if Hadiths
Despite calling these hadiths ones with a weak chain, Muslims follow them, because it doesn't mean the Hadith contradicts the Quran, and could possibly be a tradition from Mohammed.
In general, Da'if Hadiths have an 85%-90% chance of being true.
Even Mawdu (fabricated) hadiths are said to have a 65-70% chance of being true.