q=0.1
z=0 P=1.0000000
z=1 P=0.2045873
z=2 P=0.0509779
z=3 P=0.0131722
z=4 P=0.0034552
z=5 P=0.0009137
z=6 P=0.0002428
z=7 P=0.0000647
z=8 P=0.0000173
z=9 P=0.0000046
z=10 P=0.0000012
q=0.3
z=0 P=1.0000000
z=5 P=0.1773523
z=10 P=0.0416605
z=15 P=0.0101008
z=20 P=0.0024804
z=25 P=0.0006132
z=30 P=0.0001522
z=35 P=0.0000379
z=40 P=0.0000095
z=45 P=0.0000024
z=50 P=0.0000006
Solving for P less than 0.1%...
P < 0.001
q=0.10 z=5
q=0.15 z=8
q=0.20 z=11
q=0.25 z=15
q=0.30 z=24
q=0.35 z=41
q=0.40 z=89
q=0.45 z=340
12. Conclusion
We have proposed a system for electronic transactions without
relying on trust. We started with the usual framework of coins
made from digital signatures, which provides strong control of
ownership, but is incomplete without a way to prevent
double-spending. To solve this, we proposed a peer-to-peer network
using proof-of-work to record a public history of transactions
that quickly becomes computationally impractical for an attacker
to change if honest nodes control a majority of CPU power. The
network is robust in its unstructured simplicity. Nodes work all
at once with little coordination. They do not need to be
identified, since messages are not routed to any particular place
and only need to be delivered on a best effort basis. Nodes can
leave and rejoin the network at will, accepting the proof-of-work
chain as proof of what happened while they were gone. They vote
with their CPU power, expressing their acceptance of valid blocks
by working on extending them and rejecting invalid blocks by
refusing to work on them. Any needed rules and incentives can be
enforced with this consensus mechanism.