I used the phrase "probability of the second failing during the remainder of the flight is totally unaffected by that event."
EASA says the same thing, more or less, but differently;
"...objective that the catastrophic loss of thrust from independent causes is no worse than extremely improbable."
There's another problem; ETOPS planning uses the notion that the risk of the second failure - "catastrophic loss of thrust" - increases as time goes on after the first failure. That's why the longer the diversion flight time allowed for an aircraft , the lower its in-flight shut down rate has to be.
But if the second failure is entirely independent of the first, it is a random event, and as such the probability of it happening is exactly the same at any point on the time line of the diversion flight, or indeed at any time, period. The length of the diversion flight makes no difference.
There's a lack of consistency here. I have always suspected that ETOPS was driven by manufacturers wanting to build and sell big twins, and their engines, and airlines wanting to use them economically, ie long flights over water, and that this imperative has created some spurious statistical justification to allow that to happen.
But we have to remember that regardless of all that, the risks of the first failure, followed by a second one, are so low that the "catastrophic loss of thrust" will probably never kill people. None-the-less, I will continue to avoid twins on long over-water or polar flights, although I would take my chances with a survivable forced landing on the tundra or desert.