Can someone randomly guess the key to AES 256 bit encryption? Is it possible that someone can randomly input a key to AES 256 bit encryption and it is correct? What are the chances of this happening? Is it even possible or not?
There are $2^{256}$ different AES keys, the chance that you hit the one right one on first try is thus $2^{-256}=\frac1{2^{256}}$.

To put this into perspective, here's a list of events that is roughly a billion times more likely to all happen (copied from Thomas Pornin's answer on Information Security SE):


  
  The computer spontaneously catches fire during the key generation process.
  Great Britain is wiped out by a falling asteroid during the very same second.
  A rogue gorilla escaped from a zoo enters your living room and mauls you.
  You win millions of dollars at the lottery three times in a row.
  


According to his answer, these events have roughly the chances of happening $2^{-45},2^{-50},2^{-60},2^{-71}$ each, which multiplies to $2^{-226}$ if you want all four to happen which is still roughly a billion times more likely than finding a 256-bit key on first try.

Note that this assumes that the key does indeed contain 256-bit of entropy, ie it can indeed take all the $2^{256}$ different values.