I find it odd that some backgrounds give the character proficiency with tools but then do not list those tools under equipment.
For example, the Urchin:
Tool Proficiencies: Disguise kit, thieves' tools
Equipment: A small knife, a map of the city you grew up in, a pet mouse, a token to remember your parents by, a set of common clothes, and a belt pouch containing 10 gp
Does the fact the Urchin is proficient with the disguise kit and thieves tools imply she starts with them? Or is she somehow proficient with tools she does not own?
Best Answer
You have 2 choices for starting equipment. One is to take the equipment given by your class and background, which in some cases (such as the Rogue) includes tools, and in other cases (such as the Fighter) does not.
The other option is to take starting gold as given at the start of Chapter 5, and buy items from the basic equipment lists up to that amount. If you want a tool that your class and background do not provide, you can take starting gold and buy that tool. (Note that if you choose to take starting gold instead, you do not get any equipment from your class or background.)
You should consider, though, that in general the beginning of your life as an adventurer is meant to be a significant change from whatever you were before. (Such as a criminal, or urchin, and so on.) So even if you had those tools in your past, and became proficient with them during your former occupation, whatever caused you to become an adventurer might have involved leaving some possessions behind, or some things been stolen from you.
It's not so much a case of "proficient with tools she does not own" as "proficient with tools she had at some point but doesn't have now for some reason." But as I said earlier, if you want to have something that your starting equipment does not provide, you're free to take starting gold and go shopping for it.