To answer just the question in the OP, if we use SHA-512 (as it is now I think) and we assume it’s a “good” hash (its values are evenly distributed) then the chances for a collision of two blobs are 1 in 2^256 (1/115792089237316195423570985008687907853269984665640564039457584007913129639936) and that is very unlikely to happen within the lifetime of the universe. I’ll try to get the birthday-paradox answer as well; just gimme a second.
Sometimes it’s said that when a probability is smaller than the chances of getting killed by a meteorite, it’s time to move on and worry about something else 
UPDATE: based on the p2/22n+1 approximate formula from stackoverflow: the global chance for a collision with p = 160 trillion messages and SHA-512 (n = 512) is around 9.5e-127. With SHA-256, it is “only” 1.1e-49 (0.00000000000000000000000000000000000000000000000011).
The number for SHA-512, in everyday format, just for kicks: 0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000095
To spice it up even more, let’s assume a trillion times more blocks, and “only” SHA-256: the chances that the hashes of any of our 160 trillion trillion messages collide is around 0.00000000000000000000000011. I think we’re safe on this front.