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Scaling up probabilities in space

Suppose you have some location or small area, call it location A, and you have decided for this location the 1-in-100 year event for some magnitude in that area is ‘x’. That is to say, the probability of an event with magnitude exceeding ‘x’ in the next year at location A is 1/100. For clarity, I would rather state the exact definition, rather than say ‘1-in-100 year event’.

Now suppose you have a second location, call it location B, and you are worried about an event exceeding ‘x’ in the next year at either location A or location B. For simplicity suppose that ‘x’ is the 1-in-100 year event at location B as well, and suppose also that the magnitude of events at the two locations are probabilistically independent. In this case “an event exceeding ‘x’ in the next year at either A or B” is the logical complement of “no event exceeding ‘x’ in the next year at A, AND no event exceeding ‘x’ in the next year at B”; in logic this is known as De Morgan’s Law. This gives us the result:

Pr(an event exceeding ‘x’ in the next year at either A or B) = 1 – (1 – 1/100) * (1 – 1/100).

This argument generalises to any number of locations. Suppose our locations are numbered from 1 up to n, and let ‘p_i’ be the probability that the magnitude exceeds some threshold ‘x’ in the next year at location i. I will write ‘somewhere’ for ‘somewhere in the union of the n locations’. Then, assuming probabilistic independence as before,

Pr(an event exceeding ‘x’ in the next year somewhere) = 1 – (1 – p_1) * … * (1 – p_n).

If the sum of all of the p_i’s is less than about 0.1, then there is a good approximation to this value, namely

Pr(an event exceeding ‘x’ in the next year somewhere) = p_1 + … + p_n, approximately.

But don’t use this approximation if the result is more than about 0.1, use the proper formula instead.

One thing to remember is that if ‘x’ is the 1-in-100 year event for a single location, it is NOT the 1-in-100 year event for two or more locations.  Suppose that you have ten locations, and x is the 1-in-100 year event for each location, and assume probabilistic independence as before.  Then the probability of an event exceeding ‘x’ in the next year somewhere is 1/10. In other words, ‘x’ is the 1-in-10 year event over the union of the ten locations. Conversely, if you want the 1-in-100 year event over the union of the ten locations then you need to find the 1-in-1000 year event at an individual location.

These calculations all assumed that the magnitudes were probabilistically independent across locations. This was for simplicity: the probability calculus tells us exactly how to compute the probability of an event exceeding ‘x’ in the next year somewhere, for any joint distribution of the magnitudes at the locations. This is more complicated: ask your friendly statistician (who will tell you about the awesome inclusion/exclusion formula). The basic message doesn’t change, though. The probability of exceeding ‘x’ somewhere depends on the number of locations you are considering. Or, in terms of areas, the probability of exceeding ‘x’ somewhere depends on the size of the region you are considering.

Blog post by Prof. Jonathan Rougier, Professor of Statistical Science.

First blog in series here.

Second blog in series here.

Third blog in series here.

Fourth blog in series here.

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