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 (100x10¹⁰x10³) and N is the number of possible hashes (2¹⁶⁰).

This is a difficult number to calculate, but there is a good approximation: p = 1 - e^-k(k-1)/2N

But even that value is difficult to compute because of the precision needed. Here is another approximation p = k²/2N.

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-probabilities

Ok, 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".

A graphical explanation of bitcoin security http://i.imgur.com/VjtG3.jpg

-snip-
What you people think about that? This can become a big problem
for Bitcoin in the future... no?

I am a little scared