I'm pretty sure that this is the same math we played with in Re: [OT] The statistics of hashing.. I don't know how to compute the standard deviation, though. I'll have to do a bit of reading and see what I can come up with. But judging from the results from that thread, I'd expect there to be minimal overlap between two sets of 1e6 bits in 4e9 possibilities.
Update: I found my ana_2.pl script, but since I'm on a 32-bit machine, I couldn't run it with 2^32 bit vectors. (I really need to stand up a 64-bit OS and perl one day.) But I ran it with a million samples in a pair of vectors of various sizes (2^24, 2^26, 2^28, 2^30 and 2^31) and it looks like collisions shouldn't be very frequent, judging from the progression:
$ ./ana_2.pl 24 2 1000000
N=16777216, V=2, X=1000000 integral(1000000)=25166956.740071, integral
Expected collisions: 1132.74007095024
$ ./ana_2.pl 26 2 1000000
N=67108864, V=2, X=1000000 integral(1000000)=100663369.193409, integra
Expected collisions: 73.1934093236923
$ ./ana_2.pl 28 2 1000000
N=268435456, V=2, X=1000000 integral(1000000)=402653188.613027, integr
Expected collisions: 4.61302703619003
$ ./ana_2.pl 30 2 1000000
N=1073741824, V=2, X=1000000 integral(1000000)=1610612736.28892, integ
Expected collisions: 0.288918733596802
$ ./ana_2.pl 31 2 1000000
N=2147483648, V=2, X=1000000 integral(1000000)=3221225472.07225, integ
Expected collisions: 0.0722546577453613
When your only tool is a hammer, all problems look like your thumb.
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