I have a craftsman 2 cycle snow blower that starts up fine but when I turn the choke off it dies. If I leave choke on it will stay running for a while (15 or 20 minutes) then will start to die out. If I press the primer button as it is dying it will prevent it from shutting off and will continue running for another few minutes. I have fresh gas/oil mixture in it and proper ratio,I have cleaned the plug, and cleaned small air inlet on carb (small brass colored pipe), I have also cleaned carb. What's wrong and did I miss anything that I should have done?

Hi,
Just based on info given, I would say it is probably a Tecumseh engine and you missed some fuel passages while cleaning. In particular the diagonal on in the main jet that feeds the low speed circuits. A guess may be that this is your manual:
http://www.smallenginesuppliers.com/html/engine-specs/tecumseh/Tecumseh_HSK-HXL_TVS-TVXL_Engines_Service_Information.pdf
And referring to page 9 and the drawing of the main jet. That hole between and at the top of the lower threads is the diagonal hole.
Peace,
Clarke