My bathroom gets alot of hair on the floor from brushes and stuff.....and its really hard to notice where the hair actually is. So i need a way to remove it because it's kind of discusting me to go in there. I need an effective way that doesn't involve alot of spending

I would like to replace floor tiles in my bathroom. It looks like it only has plywood, mortar, and tiles (from botton to top). Can I just follow this setting to replace it or I'd better have backer board in between.