You are misusing the word "probability" in your question. 10 to the power 120 is known as the Shannon number and is a lower bound for the game-tree complexity (number of different games of chess). This has nothing to do with probability. Also this number of different games is not the same as the number of possible chess positions (which is estimated as 10^43).
If you look at the starting position in chess, each player has 20 possible moves (16 with the pawns and 4 with the knights). So the number of possible games of 1 move (2 plies, i.e. one white move and one black move) would be 20*20 = 400 already.
Now the number of possible moves increases initially because your pieces get more space. The number of possible second moves is not fixed, but will depend on the first move. So it is impossible to give an exact answer to your second question. But you can do some estimates...
If you say that at each move you have a choice between 30 moves (this is an estimate) and if you say that an average chess game lasts 40 moves (80 plies), you end up with the number of possible games equal to: 30^80=(30^2)^40 which is approximately equal to 1000^40=10^120.
This is a very rough estimate but is sufficient to show that chess is incredibly complex and is not going to be solved anytime soon.
You might also like the Numberphile episode on the topic