What happens when you visit Adobe's about flash page? (https://www.adobe.com/software/flash/about/)
It should display your flash version as well as play a short flash animation near the top of the page.
The problem also could lie in any plugins/extensions you might have installed on your browser that simply conflict with the flash player in some way. But in that case I'd have to ask if flash works on any sites you visit, or if this is specific to Youtube?
"Given billions of tries, could a spilled bottle of ink ever fall into the words of Shakespeare?"