Do characters start with the tools their background says they are proficient in?

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?