Running some results, we can see the probability drop off exponentially with z.
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
The Bitcoin whitepaper has this information which I assume is related to confirmations but this does not make any sense to me. What is Satoshi trying to say here?
He is showing an example:
q = probability the attacker finds the next block.
You get a table for q=0.1, so basically an attacker with 10% of the hashrate, and the probability of catching up with the real chain of depth
z.
He sampled the same table for q=0.3 (30% of the hashrate) as well.
The last part shows if you want this probability to be under 0.1%, with an attacker that has 10% of the hashrate, you need to wait for 5 confirmations, or if an attacker has 30% of the hashrate you need 24 confirmations and so on.