Just some odd bits plucked from the internet...
However, if one was to chose a random seed, here are the probabilities of accidentally ending up with the same keypair / address:
To obtain the same address, the result of RIPEMD-160 hashing would need to be the same. Probability of that happening is 1 in 2^160.
To obtain the same private key, it is a probability of 1 in n, where n is the order of secp256k1 and equal to FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE BAAEDCE6 AF48A03B BFD25E8C D0364141, or about 2^256.
In simple terms, if you were testing against 500k patterns at 25/Mkey a second, it'd take about 25 quadrillion years to get a few collisions. Factoring out the randomness of course, as the nature of randomness entails that it could take 2 seconds or 2 sextillion years.
Even if you put all the calculating power of the entire bitcoin network on it, it would still take 10^12+ years.
This site explains it quite nicely...
Let’s say it’s 2045 and the world population is 9 billion. In an unrealistic scenario, every single person on the earth is using Bitcoin and each person has created and used 10 million addresses; yes let’s go overboard with this: that would mean a total of 90,000,000,000,000,000 addresses spent addresses.
So, the possibility of collision, in this scenario is:
90,000,000,000,000,000 / 2^160 = 0.00000000000000000000000000000000615%
You see, 2^160 is really a very large number. Very large. Much larger than our brains can comprehend.
http://www.miguelmoreno.net/bitcoin-address-collision/Comparatively speaking, your odds of being struck by lightning are about 1 in 280,000, so you’re about 500,000,000,000,000,000,000 times more likely to be struck by lightning than to find an address within the first year. Since that’s also a big number, the odds are equivalent to being struck by lightning about 4.6 times in your lifetime.
