Combining probabilities of nuclear accidents The recent events in Japan have made me think about the following.
Nuclear Plants are usually designed to limit risk of serious accidents
to a 'design basis probability' for example, say, 10E-6/year.
This is the criteria for a single plant.
However, when there is a population of hundreds of reactors, how do we combine the individual probabilities of a serious accident ?
I know I could probably research this myself but having found this site I a sure
there is someone that will be able to answer this question quite easily.
Thanks
 A: The underlying difficulty behind the question is that situations that have been anticipated, have generally been planned for, with mitigation measures in place. Which means that the situation should not even turn into a serious accident.
The serious accidents stem from unanticipated situations. Which means that you cannot assess probabilities for them - they are your Rumsfeldian unknown unknowns.
The assumption of independence is clearly invalid - Fukushima Daiichi shows that. Nuclear plants can have common-mode failures. (i.e. more than one reactor becoming unavailable at once, due to a common cause).
Although probabilities cannot be quantitatively calculated, we can make some qualitative assertions about common-mode failures.
For example: if the plants are all built to the same design, then they are more likely to have common-mode failures (for example the known problem with pressurizer cracks in EPRs / PWRs)
If the plant sites share geographic commonalities, they are more likely to have common-mode failures: for example, if they all lie on the same earthquake fault line; or if they all rely on similar rivers within a single climatic zone for cooling (when a very dry summer can cause all such plants to be taken offline).
A: As commentators pointed out, this has the very strong independency assumption.
Let the probability that a plant blows up be $p$. Then the probability that a plant does not blow up is $1-p$. Then the probability that $n$ plants do not blow up is $(1-p)^n$. The expected number of plants blown up per year is $np$.
In case you're interested: binomial distribution.
A: To answer the pure probabilistic question that J Presley presented, using bayer's notation (p=probability of an item failing), the the probability of at least one element failing is 1-P(none fail)= 1-(1-p)^n. This type of calculation is common in system reliability where a bunch of components are linked in parallel, so that the system continues to function if at least one component is functioning.
You can still use this formula even if each plant item has a different failure probability (p_i). The formula would then be 1- (1-p_1)(1-p_2)...(1-p_n).
