You have some strong signal about something, but it's usually hard to take anything as gospel. Instead, you take it as a datapoint and investigate further.
You can try treating every alarm as a false alarm (because, historically, nuclear wars have never been started), but still investigate what is happening. The alarm isn't useless, just not decisive.
P(attack | alarm) >= P(attack)
Unless the alarm is literally random, or even anti-correlated with being attacked. The alarm is still useful.
Remember that the penalty for missing a true alarm is possibly even worse that reacting to a false alarm.
That depends on the scenario. The US for a few years advocated a "total" nuclear response to any use of nukes against western nations. The theory went that any limited nuclear exchange would inevitably escalate. So to win you want to fire everything in the first round. In such scenarios ignoring a true alarm, not triggering the planned world-ending response, may be the safer choice. One nuke in the US, and zero in the USSR, may be the better outcome than hundreds landing on each side. (See "The Sum of All Fears" for how that plays out.)
If you have an alarm, there is some probability that it's a false one. If the US were to respond to a false alarm by say, nuking Russia, they could expect a 100% chance that Russia would retaliate. The retaliation would be extremely severe if either the US or Russia gets nuked. You have to think that in addition to all ground-based silos (some of which would definitely survive a nuclear strike), there are stealth subs with nuclear capabilities going around. Even if the US or Russia "lost" a nuclear war, these subs would be there to exercise some kind of vengeance, and make sure that the other side is destroyed as well.
If some alarm goes off and there is some chance that nukes are going to hit the US, the safest course of action is probably to prepare to retaliate, but wait as long as possible to confirm that there is an incoming nuclear strike. This might even mean waiting to confirm that some US targets have been hit before retaliating. You also should be pretty god-damn sure you know who initiated that strike before you retaliate. China, for instance, has a limited number of ICBMs. They couldn't destroy the US even if they wanted to. Russia, on the other hand, could nuke the entire world multiple times.
Unless the attack successfully disables their response. I don't think it makes sense to rely on that, but it's definitely a point that has factored into various calculations around this issue over the years.
Unilateral nuclear disarmament is a solution that seems to be rarely discussed.
The real reason so many countries have or want to have nuclear weapons is not to ever use them or even so that they serve as a deterrent against a largely hypothetical enemy; it's as a badge of "greatness".
That's why rational arguments against nuclear weapons or which attempt to describe the utter craziness of it all, have so little impact.
I think P(attack | alarm) > P(attack) does not by itself imply that paying attention to alarms is the right thing to do.
> Remember that the penalty for missing a true alarm is possibly even worse that reacting to a false alarm.
The outcome of reacting to a false alarm is exactly the same as the outcome of reacting to a true alarm, and, arguably, the same as the outcome of not reacting or reacting only partially—it's all -∞ basically.
It must be very interesting how this fact in included on strategies.
And you never want to launch all your missiles, because then you have no bargaining power. So any nuclear conflict is likely to involve several exchanges.