Determining the probability of multiple rare events happening on the same day

Question

I earned an ace in my graduate statistics course. Nonetheless, I know when I'm treading in deep waters and need the assistance of a professional. Seeking a statistical guru (or wizard perhaps) that can help me with a probability question for my current book project. I am trying to estimate the probability of six large to massively large, natural wildfires happening within the same general area (within 250 miles) on the same day in the year 1871. Five of the fires were massive (>500K acres) and the remaining fire was large (>2000 acres).

Assumptions:

I made an arbitrary assumption that the "fire season" was six months, or 182 days long. I'm open to using a 365 day fire season but this seemed more reasonable.

I used 2018 US wildfire statistics as a premise or proxy for estimating the probability of a wildfire of a given size. In 2018 there were 58,083 wildfires in the US. 11 of those 2018 wildfires were massive (>500K acres). Two percent of the wildfires in 2018 were large. I'm making a reasonable assumption that the range in size of wildfires in 2018 is equal to the range of size of wildfires in 1871. Therefore the odds of a wildfire being a massive wildfire would equal 11 / 58,083 = 0.00019. The odds of a large wildfire occurring would be 2% of all fires.

Using these reasonable (tell me if you disagree) assumptions and probability proxies, how would you calculate the probability of these six wildfires (5 massive; one large) happening on the same day?

Advanced twist to the problem for a statistical genius:

Q. The six wildfires in the problem were located within a great circle with a diameter of 250 miles. When you take into consideration the number of great circles with a diameter of 250 miles that can fit into the land area of United States (I don't know that number), how would that information change the probability of occurrence?

All significant contributors providing a lucid, accurate and outstanding answer to my question will be duly referenced and given the appropriate honors, platitudes, thanks and recognition deserved in the Preface of my future, best-selling, Pulitzer Prize winning book (I can dream can't I?). Provide a detailed explanation (please) in the simplest statistical terms that an aging - but not over the hill yet - Baby Boomer with three degrees can comprehend and you will be my hero for life. Thank you for your consideration.

Namaste.

Eric

Answer 1

"occurring on the same day" suggests using the Poisson distribution, at the heart of which is the

# of occurrences in a given time frame. So ideally you would need to know what small,tiny fraction

of a fire occurs per day.

You have given 58+ fires in 189 days. That's lambda = 0.5532110091743119266.... which will have to server as the parameter LAMBDA, the average rate of occurrence.

So per the Poisson on model , the said lambda, k=6 fires, lambda^6 / (exp(lambda))(6!)

This calculation gives you a probably of order -3 , specifically 1.85175570240888... * 10^(-5)

However, you are also requiring the fires be with the same 125 mile radius ( as you have given a diameter of 250 miles). Assuming the location of where the fire starts is an INDEPENDENT event ( we are treading on dangerous waters here.....) and that each location has an equal chance of having a fire started ( which is BINOMIAL distribution) then based on the given measurements:

The land area of the continental United states is 3.797 million square miles.

So now we have (125 / 3797000)^6 which will decrease the Poisson probability by a whopping

factor of order -25, namely 1.272965 * 10^(-27)

Multiplying said probabilities in bold brings the likelihood down to 2.3572201977169199292 x 10 ^(-32),

which is of order -30.

Finally, you are requiring these fires be on sufficient strength, magnitude, and intensity.

The combined probabilities you have given are 2% + 0.00019 = 2.019%. Again, assuming

independence, we have (2.019%)^6 gives an order -9 approximation of 0.000000000067735745289+

= 6.7735745289 * 10^(-11)

Multiplying naively all of these assumed independent events, the final product is

8.6225233001811885 * 10^(-39) = 0.0000000000000000000000000000000000000086225233001811885%

There is my bonfide attempt backed up by the good old fashioned college try.

Godspeed