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July 27th, 2019, 02:22 PM  #1 
Newbie Joined: Jul 2019 From: United States Posts: 2 Thanks: 0  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. Assumptions:  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 
July 28th, 2019, 12:31 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,823 Thanks: 723 
The statement and the data need some geographical context. Wildfires are not uniformly distributed over the entire country.

July 28th, 2019, 01:37 PM  #3 
Newbie Joined: Jul 2019 From: United States Posts: 2 Thanks: 0 
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. 
July 28th, 2019, 02:16 PM  #4 
Senior Member Joined: Jun 2019 From: USA Posts: 213 Thanks: 90 
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 fireprone areas are more vigilant under certain meteorological conditions (drought, high temperature, …). I don't think you can ignore these effects or the crosscorrelation 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. 
July 29th, 2019, 05:36 AM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,683 Thanks: 2664 Math Focus: Mainly analysis and algebra 
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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day, events, happening, multiple, poisson probability, probability, rare 
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