After reading Tensibai's excellent answer, I realised I used to be able to calculate this for network analysis purposes. I dug out my copy of High Availability Network Fundamentals by Chris Oggerino and had a crack at working this out from, not quite first principals.
Taking my serial example directly out of Tensibai's answer is simply a case of multiplying the probability of each component being available by the other:
So
99.95% * 99.95% = 99.9%
Calculating it in parallel is a little more complicated as we do need to consider what the percentage unavailability will be:
The calculation is done as follows:
Multiply the unavailability of the two regions together.
0.1% * 0.1% = 0.0001%
Convert that back to availability
100% - 0.0001% = 99.9999%
Multiply the Traffic Manager availability by the availability of the two regions.
99.99% * 99.9999% = 99.9899%
The result is the whole system availability.
99.9899% is close to 99.99%
I ended up using Excel to perform the calculations, here is the values:
... and the formulas ...
