I have a cluster of eleven machines each machine having five hard disks. I am using Hadoop distributed file system (HDFS) which takes each file I upload to the cluster and cut it into blocks and then replicate each block three times on different machines. This means I can lose two from the eleven machines without losing any files because blocks can be restored from the third replicas on some other machines.
Recently I moved my cluster and three hard disks from three different machines were damaged while shipping which means there was a chance that some blocks could have been lost if they were replicated on those exact three disks that were damaged. However that didn't happen and no data was lost.
I told my colleagues that I am lucky this didn't happen but then I started to think whether I am really lucky or maybe the chance of this happening was low to begin with.
I have about 120,000 blocks in my file system. So here is my thinking:
Given a block, there are $\binom {11}{3} = 165$ ways to choose three machines to replicate the block. Once we choose the three machines then there are $5 \times 5 \times 5=125$ ways to choose three disks. So in total there are $165 \times 125=20625$ ways to choose three disks to replicate the block.
So the probability of a block going to the three corrupted disks is $\frac{1}{20625} = 0.00004848484 $
What would be the next step from here to find the probability that at least one block form the 120,000 blocks could have been replicated on the three corrupted disks?
I would say that the probability a particular block is on all three damaged disks is $\frac{3}{55}\times \frac{2}{50} \times \frac{1}{45}$ which gives the same $\dfrac{1}{20625}$ you found.
So the probability it is not is $\dfrac{20624}{20625}$.
Assuming blocks are distributed independently of each other (which may be almost the case), that would make the probability that none of the blocks are on all three damaged disks $\left(\dfrac{20624}{20625}\right)^{120000} \approx 0.003$ (this is also about $e^{-120000/20625} )$
So either you were very lucky (though not impossibly so), or something else happened.