How fast of an Internet connection do I need to play Dota 2? I have 2mbps internet speed, do you think I can play Dota 2 smoothly?
Your internet speed (2mbps), should be fine. The most important thing for playing "smoothly" is ping. You can use this test to find your ping between servers around the world. You should pick the server based on the region you're most likely to play on
US West Seattle, WA, USA
US East Sterling, VA, USA
Europe West Luxembourg
Europe East Vienna, Austria
SE Asia Singapore
South America São Paulo, Brazil
Russia Stockholm, Sweden
Australia Sydney, Australia
Anything below 50 is ideal. Below 150 is probably playable. 200 becomes a little shifty. If you get ping "spikes" (when your ping rapidly rises between say 30 and 150), quite frequently, this can throw you off your play, as latency kicks in, different actions take different amounts of time to process, and the game feels sluggish. The only way to determine if this happens is by playing the game. It's a free to play game regardless, so you lose little by installing it.
My friend had a 2Mbps connection at his house once. He could play DotA2 reasonably well with about 50 ping when ONLY HE was connected to the internet and NO ONE ELSE was using it.
Whenever his roommate connected to the WiFi and tried to load a page his game would lag. If his roommate tried to watch a YouTube video DotA would auto disconnect.
If it is a dedicated 2Mbps connection only for you and noone else will be using it you should be able to play properly... otherwise you can expect some disturbance.