That's assuming Josh needs to.
IME, depending of course upon what one's bank uses, you can buy something from eBay and it will say the name of the store and some internal string of gobbledegook; but it won't give any hint as to what you bought.
Other secondhand markets may do the same type of thing, naming the store not the item.
This also assumes that Anna will look into or care about such a purchase; and that Josh is wedded to an expensive desktop for pursuing his perversions. If he opts for something like a Kindle Fire from a secondhand dealer, he could easily spend less than $100 on a second device.
correct. I meant more along the lines of her physically noticing a new laptop/tablet/whatever; it’s easier to just set up a partition on an existing device instead of buying a new one.