If you become a special adviser to the president, would you point to the president such an impossible coincidence and advice your president to take the July 4th off and relax avoiding the suspicious spell. Or would you explain such coincidence in some other way?

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## 8 comments:

If I were special advisor to the president, I'd tell him to relax and enjoy July 4th because it's a national holiday, and we should all have a BBQ and go to fireworks and enjoy the day.

But I wouldn't tell them to relax because of some superstition about a curse with the date. If there had been a curse, it seems to be long-since broken at this point, with no president since the 5th one dying on that day.

That said, the odds of three of the first five presidents randomly dying on the same day are pretty slim. So, if I had been an advisor to John Quincy Adams or Andrew Jackson, I might have had a different take on it!

Assuming that all three death dates are independent then the probability that the second president has the same death day as the first is 1/365. That's because whatever that death day happens to be (July 4th) that day is just as likely to be the death day of the second president.

Similarly the probability that the third president has this deatth day is also 1/365. So the probability that all three have this death day is 1/365 times 1/365 which is equal to 1/133225.

Now Adams and Jefferson were competing with each other so maybe there is a factor of "hanging on" but also "allowing oneself to die" and the 4th of July had a psychological effect on this "decision".

The statistical probability is also a function of age and health. Climate, finance, humidity,pollution..are environmental and economical factors that could be taken into account and influencing both health and age.

The biopsychosocial model of health is an important factor of remaining alive.

Now, other factors could be taken into consideration and a book could be written. Consideraring birth date is one of these factors.

anne-marie

There is a historical footnote that could be noted.

page 669 of this reference.

http://books.google.ca/books?id=hCAjgs4mmQ4C&pg=PA659&lpg=PA659&dq=death+james+madison+medication&source=bl&ots=eb-BFd6Oiy&sig=Wt0FAS0Ijk3_k_llA8vBcZMkfxg&hl=en&ei=NZKPTrSaDOGHsgL79uGYAQ&sa=X&oi=book_result&ct=result&resnum=6&sqi=2&ved=0CD8Q6AEwBQ#v=onepage&q&f=false

James Madison refused herbal medications that would have forstalled his death for another 6 days which would have made him the fourth president to die on that momentous date.

I know that this is a math column but I'm sure that Adams, Jefferson and Monroe might have tried to extend their lives to die on July 4th, though I can find no reference for it. I am sort of implying that it is possible that Adams, Jefferson and Monroe thought highly of the idea of dying on July 4th thus making it less of a coincidence.

An advisor might point this out.

I realize that this is a math column and we are being asked to show how weird the coincidence is.

If you take three people out of a group of n what is the probability of them having the same (given day) birthday?

The general method is (for no specific date)

1- 365/365 * 364/365 * 363/365 = 0.9973 (which is almost certainty) that all three will be different. The fact that you must choose 1 certain day makes the probability even higher that there will be 3 death days the same.

And yet, here are 3 who died on July 4; 2(Gerald Ford and Harry S Truman who died on December 26) and 2 others (Filmore and Taft) died on March 8th. The importance of December 26 is obvious (1 day after Christmas). March 8 is really the only truly random day of agreement. It is almost certain that amoungst 39 presidents there would be 2 who had the same death date.

The presidents beat the odds by having 3 of 44 die on the same day. But the three who died on July 4 were also Founding Fathers, so the day would have more import to them than to anyone since. My guess is that they would have "stayed alive" until such an important anniversary, as people sometimes do, or had an elevated stress level on that anniversary. The day would have been much more personally important to them than to anyone since, and I suspect that modern presidents would not have to worry more on July 4 than any other day.

But that is not a mathematical answer...

Simplifying assumption: no leap years :-).

July 4th as a specific date is indeed a pretty unlikely as a date of death for three presidents out of 40 that had died. To be precise, it's (C(40,3)*365^37)/365^40 = C(40,3)/365^3=0.00020317878139, which is pretty slim (one out of almost 5000 chance). So as a special advisor, maybe I will be wary of this date given that it has symbolic meaning for the whole country.

On the other hand, what are the chances of any three presidents dying on the same date, but not necessarily on July 4th? Let's assume that each triple of presidents out of 40 having the same date of death is an independent event. Each triple’s probability of NOT dying on the same date is 1-(1/365^2), and combined probability of not a single triple with the same death date would be that number to the degree of all possible triples, i.e. (1-(1/365^2))^9880. So the approximate probability of at least one triple is 1-(1-(1/365^2))^9880, which is about 1 in 14. That seems not so outrageous. Exact probability may be even higher given that we are not dealing with an independent event in each triple, but I don’t have time to work it out. This is similar to the famous “Birthday Problem” :-).

You all provided so informative and interesting answers that each of you deserves a puzzle point - Kim, anne-marie, Jerome, Bean and Ilya.

One simple thing that some of you mentioned is that although we had 44 presidents, some of them are still alive and some held the post twice. So, we are talking about 38 dead presidents according to this chilling description.

If we would be advising to the 6th, 7th or 8th president, then as Kim mentioned death of the 3 out of first 5 is extremely suspicious. However, we may only be lucky to advise to the current president and up. So, out of the 38 dead presidents, 3 died on the same date. What are the odds?

As Ilya mentioned this is indeed very similar to the birthday paradox and as explained in this paradox odds of two people out of 38 sharing a birth or a death date are around 85%.

What are the chances that at least one of the remaining 36 presidents has the same death date? I think it is 0.85 x 36/365 = 0.084 - around 8%, rather close to Ilya's answer and not shockingly outrageous.

i dont know i tano wokno as my baby would say it

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