Here's the fundamental problem with ETOPs as I see it...
Say that you'd expect a single engine to fail (not necessarily catastrophically) around once every 30,000 hours of flight. Lets call the probability of single engine failure within an hour of flight Psef
So Psef = 3.33 x 10^-5
Is the probability of double engine failure (Pdef) simply given by…
Pdef = Psef x Psef = 1.11 x 10^-9 (roughly 1 in a billion).
No!
Why not? Because the events are not independent!
Psef can be calculated from the history of a particular engine operating under normal conditions. Sufficient hours of operation have been accumulated under normal operating conditions to indicate that an estimate of Psef = 3.33 x 10^-5 is not unreasonable.
In contrast, comparatively few hours of single engine flight time have been accumulated under the abnormal operating conditions likely to be associated with an ETOPs diversion. Consequently, we simply do not know just how likely it is that the second engine will fail under these conditions.
Given that the first engine has failed it is likely that the probability of the second engine failing is substantially greater than Psef so the probability of dual failure is substantially greater than Psef x Psef.
Note that ETOPs regulations do acknowledge this problem and try to make these events as independent as possible. For each engine - different overhaul times, different mechanics etc. However, these precautions don't deal with the fact that an engine operating alone during an ETOPs diversion is not operating under the same conditions as one of a fully functional pair during normal flight.