How do you calculate the compound Service Level Agreement (SLA) for cloud services?
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 ...