My Math Forum Probability of multiple rare events happening on the same day

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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 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.
 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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