## Probability of nuclear accidents in a country with 19 nuclear reactors

By Thomas Rose, on 28 March 2011

There are a lot of studies on the probability of accidents in a nuclear power plant. As far as I understand they use methods of risk analysis to calculate the failure probability of the nuclear reactor.

Here I tried a very simple empirical approach: We know the number of nuclear power reactors in the world, we know (probably) the number of severe accidents up to now, so we can calculate the empirical failure probability of a single reactor per year. Thus we are able to calculate the probability that no reactor in the world, in UK or in another country, will have an accident within the next 5, 10 or 20 years.Or that at least on reactor will fail severely. This can be done by using the Poisson distribution.

Up to now there are at least 4 reactor accidents on INES scale 5 or more. Chernobyl (1986) is the only one on level 7, Three Miles Island(1979), Windscale (1957) are on level 5. Also the present Fukushima accident (or accidents?) is level 5, at least at the moment (27.03.2011). On level 5 there are some more accidents and on level 6 is only one, but they were in other nuclear facilities, not in power reactors. One could argue that Windscale was not a civil but a military reactor, but then in Fukushima there is probably more than one reactor involved. So the number of 4 severe accidents seems quite reasonable.

The number of nuclear reactors worldwide increased drastically from 1955 until 1988, from which date the number is nearly constant. Up to the Fukushima accident there were 443 reactors operating worldwide.

By a simple graphical piecewise interpolation of the number of reactors per year a total of 15.000 reactoryears can be estimated. This crude number should be sufficient for the present purpose.

So the probalilty for one severe accident per reactoryear (ry)is

q=4acc/15.000ry

If there are N reactors in operation, the Poisson distribution gives the probability for x severe accidents within the next y years. In order to apply the Poisson distribution the expected mean number of accidents m within this time has to be estimated:

m=q*N*y

Then the probality to have x accidents when we expect a mean value of m accidents is given by

p(x)=(m^x/x!)*exp(-m)

Thus the probality for no accident is (x=0)

P(0)=exp(-m)

and the probality for at least one accident is

p(x≥1)=1-exp(-m)

Regarding the worldwide situation for the next 20 years, the number of reactors is 443, we expect an average number of severe accidents

m=(4acc/15.000ry)*443r*20y=2.34acc

so 2.34 accidents within any period of 20 years somewhere in the world. The probability for one or more severe accidents worldwide is

p(x≥1)=1-exp(-2.34)=90.36%

How is the situation for a single country? We simply have to count the number of reactors within this country and calculate the respective reactoryears.

World UK US D

reactors 443 19 104 17

reactoryears 8860 380 2080 340

mean # acc 2.34 0.100 0.549 0.089

p(≥1 90.36% 9.55% 42.26% 8.59%

On the average more than 2 accidents are expected worldwide, the probality for at least one accident ist 90% worldwide, more than 9% for the UK and more than 40% for the US.

Do you think these estimations are reasonable? Do you think a 9% probability for a Chernobyl or Fukushima accident in the UK within the next 20 years is acceptable?

I am looking forward to your comments.

440+ reactors worldwide, yet the earth’s global ecology is at risk from one single “incident.”

~IMHO, the human race is not technologically advanced enough to play God with nuclear energy. Prayers for all the Japanese people. (and all of earths inhabitants)

[...] calculated globally, but can also be done for the Indian scenario. Following the methodology given here, the probability of a severe nuclear accident (INES greater than 5) is got by assuming, [...]

That is very scary , a 9% probabiity over 20 years for the UK. does that mean a 45% probability over 100 years and a 90% probability over 200 years ?

I am very poor at maths can someone please answer the question that Dani posed , does this mean a 45% probability over 100 years ?

Thank you

[...] Anglesey’s nukes are unlikely, on balance, to be built. Capital costs determine their economic viability and capital is in – well, let’s call it short supply. They nonetheless remain a looming elephant in the region’s room. (Curiously, the only discussions that crop up during my visit concern what to do about roosting bats displaced by site clearance – and the decision to build a third bridge off the island for people to escape if a nuke blows.). [...]

These estimations are really scary. When i take the example of Fukushima that has suffered most from nuclear accidents and has undergone enormous damage lately, Japan shall meet all its efforts to put a stop to all that in order to avoid any possible nuclear accidents . The world is still threatened by severe nuclear accidents. .. While hoping that these are only probabilities, and pray that it does not happen.

How can it be calculated for a country which intends to build a nuclear reactor in some years to come?

[...] University College London has a good post examining the statistics of nuclear accidents in countries depending on their numbers of nuclear [...]

[...] University College London has a good post examining the statistics of nuclear accidents in countries depending on their numbers of nuclear [...]

These statics are telling that human define there end.These plant are in developed country so if these nuclear power reactors have any problem 50%- 70% of the part will explode and as I know rest part of the world is already suffering from food inefficiency.I my suggestion people have to find other solution or make something that control these nuclear power reactors in any case.

As the states “a 9% probabiity over 20 years for the UK. does that mean a 45% probability over 100 years”.

If it is correct what about for the US. Does that mean a 210% probability over 100 years? As a 42% probabiity over 20 years for the US. Please correct me if i am wrong.

I am very poor in maths but,I really worried after reading this post. Is there any way to avoid the accident? If world is in danger for this nuclear reactor, how government can seat idealy?

I am so scared after reading these figures. I have also read some where about nuclear meltdown that If a single nuclear meltdown were to occur in Western Europe, around 28 million people on average would be affected by contamination of more than 40 kilobecquerels per square meter. This figure is even higher in southern Asia, due to the dense populations. A major nuclear accident there would affect around 34 million people, while in the eastern USA and in East Asia this would be 14 to 21 million people.

There is a lot of maths you used here and I don’t like maths. But I can say these things are very scary and we should do something in this matter and we should be prepared for these kind of accidents and these accidents can be big disasters so just be prepared.

Definitely these statistics are scary and these kind of things are very dangerous but tell me can anyone live without electricity now days? Can anyone live without phones now days? Can anyone travel hundreds and thousands of miles by foot?

Now days we all are habitual of these luxurious things and can’t live without them and all of these need power to work. As everyone know there are very few power sources available so we need some alternatives and nuclear power plants are very good example of these alternatives. I know there is a big risk associated with it but I heard people die slipping in bathroom that doesn’t mean we shouldn’t go to bathroom. It means that we need some extra precautions and safty.

[...] The University College London has a good piece examining the statistics of nuclear accidents in countries depending on their numbers of nuclear reactors. It is a really good introduction to the difficulties of probabilities in stochastic systems and it concludes that in a world of 443 reactors (19 in the UK, 58 in France and 104 in the US) we can expect 2.34 accidents within any period of 20 years somewhere in the world. The probability for one or more severe accidents worldwide is 90.36%. Brought back to a single country, the calculation concludes that there is a 9% probability for a Chernobyl or Fukushima accident in the UK within the next 20 years… [...]

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good article, thanks.