I’ve routinely found the standard Adobe Flash player updater for Mac (OS X/macOS) fails right at the end, without explanation.
However there is a version that always works. Go to:
https://helpx.adobe.com/flash-player/kb/installation-problems-flash-player-mac.html
… and choose Flash player for Safari and Firefox – NPAPI – the download URL is normally:
https://fpdownload.macromedia.com/pub/flashplayer/latest/help/install_flash_player_osx.dmg
The opening dialog box should look a bit different – like this:
Opera users
Use this version (last tested 10 Feb 2017):
https://fpdownload.macromedia.com/pub/flashplayer/latest/help/install_flash_player_osx_ppapi.dmg
Firefox plugin checker
This is a very fast and handy way of seeing if your copy of Flash or Silverlight is up to date.