Given your example of 1 billion users at 10 addresses each:
There are 2^160 or about 1,460,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 possible addresses
In your scenario, 1,000,000,000 people are using 10 addresses each for a total of 10,000,000,000 possible addresses
10,000,000,000 / 2^160 should yield the probability of a collision occurring
10,000,000,000 / 2^160 = 0.00000000000000000000000000000000000000684
So the chances of a collision occurring in your scenario are approximately 0.000000000000000000000000000000000000684%
See why we don't consider collisions an issue?
The probability of a collision is found by a standard formula: p = 1 - k! / N
k-1(N-k)!, where k is the number of hashes generated (100x10
10x10
3) and N is the number of possible hashes (2
160).
This is a difficult number to calculate, but there is a good approximation: p = 1 - e
-k(k-1)/2NBut even that value is difficult to compute because of the precision needed. Here is another approximation p = k
2/2N.
So the answer is that the probability of at least one collision is approximately 7x10
-19 or 0.00000000000000007%
See:
http://preshing.com/20110504/hash-collision-probabilitiesOk, new data, will recalc everything:
- probability of getting struck by lightning in any given year: 1/280000.
- probability of taking a shit at any given point in time: 1/(60*24) = 1/1440 (assuming you take a crap every day and the actual process takes 1 minute)
- probability of getting struck by lightning while taking a crap in any given year: 1/(280000*1440) = 1/1.47E11 = 2.48E-9
- probability of taking a crap while being in a situation where being struck by lightning can actually occur = 1/1440 = 0.25 = 1.74E-4
- probability of finding a collision: 1E-65
- getting hit by lightning while taking a crap for how many years in a row is equally probable as finding a collision: log(1E-65) / log(1.74E-4) = 17.3
is my math roughly correct now?
If so, I can say: "Finding a collision is about as likely as being struck by lightning while taking a crap every year for 17 years in a row".
-snip-
What you people think about that? This can become a big problem
for Bitcoin in the future... no?
I am a little scared
There is nothing to be scared of. Lucky guy get others' money. Isn't it cool?