HazelNuts39:
the probability of a second failure that results in the need to use the battery, and the probability that the battery will then have failed, increases with the remaining flight time.
I don't think you want to cut probabilities that close when it comes to critical systems. The difference between ETOPS/non-ETOPS range or flight time is on the order of 3:1 or 4:1.
If a combination of events cannot be shown to be extremely improbable (< 10^-9 per flight hour), then some backup system needs to be provided. The current LiON batteries' service history is several orders of magnitude less than what Boeing predicted. Not just the probability of fire, but of the battery functioning. That is significantly worse than 4 times.
Walnut:
An a/c can not dispatch without a servicable main battery, there are lots of vital services eg discharge of fire bottles etc where battery pwr is needed.
I said in an earlier posting that the removal of circa 150 batteries to date is very very significant. Only 2 caught fire so the rest must have been to under performing (low charge)??
That's the key. Dispatch with a serviceable main battery. And then that battery has to remain serviceable for the duration of the flight. The two fires plus removals puts the battery at such a high probability of failure that the battery system appears to need its own backup.