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July 27th, 2019, 02:22 PM   #1
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Probability of multiple rare events happening on the same day

I seek a statistics guru to help me with a complex probability problem that is part of my current book project. To the outstanding responder(s), I will generously sing your praises and reference your name in my book if the response is robust and the methodology clearly articulated.

Problem statement:
Estimate the probability of six large to massive wildfires happening within a 125 mile radius on the same day in 1871.
- The six events are independent.
- The "fire season" or days when wildfires are likely, is 182 consecutive days.
- The number and distribution of fires by size in 2018 is considered identical to wildfires in the US in 1871.

Background Data:
I am using the 2018 US Wildfire statistics as a premise or proxy for estimating the probability of a wildfire of a given size occurring. There were 58083 wildfires in in the US in 2018. Two percent of those wildfires were large. Only 11 of those 2018 wildfires were massive (i.e., >500K acres).
In 1871 six wildfires - 5 massive and one large - initiated on the same day. All six fires started within a radius of 125 miles. Massive wildfires are quite rare. Therefore what is the probability that 5 massive fires and one large fire would all start on the same day within a relatively tiny geographic area?

I suspect that this is a Poisson probability problem. I have one graduate statistics course on my resume but I am wise enough to know when to call in the cavalry (that's you!). Please outline your approach clearly without omitting steps. Please question my assumptions and make suggestions if you think a different approach is valid. I look forward to seeing your response. Namaste, Chinchaga
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July 28th, 2019, 12:31 PM   #2
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The statement and the data need some geographical context. Wildfires are not uniformly distributed over the entire country.
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July 28th, 2019, 01:37 PM   #3
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Thanks. Great Lakes Region of the US. Another positive approach may be to forget about the geographic location issue and focus on determining the probability of 5 massive fires and one large fire all starting on the same day within the annual fire season. Here is the most important aspect of the problem that I wish to solve: If, on average, there are only 11 massive fires occurring in a typical year, what is the probability that 5 will start on the same day? Don't get bogged down in the geography issue. Hope that helps.

Last edited by Chinchaga; July 28th, 2019 at 01:43 PM.
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July 28th, 2019, 02:16 PM   #4
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I question your first assumption. Just because "fire season" is six months long doesn't mean the probability of a wildfire is equal during each of those days. We all know people in fire-prone areas are more vigilant under certain meteorological conditions (drought, high temperature, …). I don't think you can ignore these effects or the cross-correlation coefficients.

For that matter, I would do as much research as possible on your third assumption, too. Changes in vegetation, population density, and technology among other things may have had a significant impact on wildfire probability over a century and a half.
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July 29th, 2019, 05:36 AM   #5
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Before providing any bald statement of probability, I would want to know to what use you intend to put it. Statistics without context are meaningless to those who know about statistics and easily convince those who don't.
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