How could 3 reactors fail at the same time, when the manufacturer estimated the probability of core damage as 1 event in 300,000 years? If these were independent events, then their likelihood of simultaneous occurrence would be (1/300,000)³ =  once in 27,000,000,000,000,000 years (27 million-billion years, over a million times the age of the universe). But obviously these events weren't independent -- they were linked by the earthquake and tsunami.

The high estimated reliability of the reactor comes from an assumption of independence of the backup systems. These reactors absolutely require uninterrupted cooling, as the radioactive decay of the fuel generates many megawatts of heat for many days. To achieve this safely, redundant pumps, power sources, and coolant sources are incorporated in the reactor design. For example, the primary power source comes from the reactor's own generator. Independent backup power comes from the electric grid. Third tier sources are diesel generators -- instead of one, there maybe two or three. Let's say the probability of a generator failing to start is 1 every 1000 attempts. Then 3 independent generators should have a probability of all 3 of them failing to start of (1/1000)³ = 1 in a billion attempts. So each generator is big enough to cool the reactor by itself, and the likelihood of all 3 generators failing should be miniscule -- if they were truly independent. And as a final redundant backup, there are batteries, that can power the pumps (for 8 hours in the Japanese reactors, just 4 hours in many American reactors of the same design). That was considered enough time to repair at least one generator.

The fallacy of statistical independence in a nuclear power plant should be readily apparent after the accidents in Japan. The ~20' tsunami swept over the 16' tsunami walls, destroying the fuel tanks and flooding all the generators or their electrical connections. They couldn't be repaired in 8 hours because the basements they were built in had filled with sea water that could not be removed with on-site tools. So a single event, a tsunami (which is not so uncommon in Japan) correlated (co-related) otherwise independent elements. But it doesn't take something as big as a tsunami to do this. Consider human error -- what if an absent minded mechanic serviced all 3 generators and replaced the oil with antifreeze. Such an error (would you call this one error or 3?) would have likewise coupled the 3 generators, rendering them all useless and leaving the plant in great peril, if it shut down and the grid failed (it would then be wholly dependent on battery power).

There are a plethora of coupling scenarios, where the low probability of independent failures is a myth, and a single event would have exponentially higher probability of dire consequences. Many of these dependencies are strong enough to to increase the probability of coupled events. If one reactor emits radioactivity forcing abandonment of others, then their probabilities of meltdown or criticality accident would dramatically increase. So coupling doesn't just change the math by no longer multiplying independent probabilities, it can also fundamentally change individual probabilities in non-linear ways.

At Fukushima, these correlations did just that. Surely the facility had an evacuation plan, that depended on trains, roads, and infrastructure that was likely damaged or destroyed by the earthquake and tsunami. As a reactor malfunctions, it requires more human resources to manage and regain control. We can presume that emergency plans would use workers from adjacent (functioning) reactors to support the emergency -- and those emergency plans likely assumed only one reactor would fail. But with 3 simultaneous reactors loosing cooling (as well as cooling losses in the spent fuel ponds), the staff may have been overwhelmed. Did focus on the reactors lead the exhausted staff to fail to anticipate overheating and hydrogen explosions from the spent fuel pools?

Since the earliest days of nuclear power, there has been a vocal opposition arguing that the industry-stated risks are underestimated (e.g., Union of Concerned Scientists). They have not just advocated for "no nukes", but argued that better regulation, fines, and improvements can dramatically reduce the risks to more acceptable levels. For example, they report how our nearby Diablo Canyon reactor operated for 18 months in 2008-9 with a non-functional emergency core cooling system (ECCS), followed by very minor sanctions.  The likelihood of reactor disasters are indeed low but finite, and the consequences are incredibly high. So the amortized cost of an accident is approximately zero times infinity -- very difficult to determine a quantitative consensus. For years the Fukushima plants provided profitable and clean electricity, but now its payback. An old man asked a reporter, "how will I take care of the graves of my loved ones?". Farmers may have to abandon fields for hundreds of years. These are also difficult costs to quantify. Those who pay the highest external costs of nuclear power are not the same as those who who reap the majority of the benefits of it.

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