There is more blood in this bath than you can shake a zombie brain covered bat at and it's going to get worse. Do zombies even bleed? I've always wondered that.
It depends on what kind of zombie, Mac. If it's a traditional, undead zombie, then no, they don't bleed because their blood has already coagulated. If it's a disease-ridden zombie that's technically still alive, a la 28 Days Later, then yes, they do.