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Mike Humphrey[_2_] Mike Humphrey[_2_] is offline
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Default OT: Latering thinking puzzle "Why do more peoplre die on theirbithday than any other day?"

On Sat, 12 Jun 2021 21:48:44 +0100, NY wrote:
"Which leads to the rather surprising statement that, assuming you are
alive right now, you're always more likely to die today than any later
day (the probability of dying on any past day being zero, of course)."

Could you go over that bit again. I don't really follow your reasoning.
I would have thought the probability of dying on any given day will
for each successive day, once you get past a certain age. And even
before that age-related effect kicks in, why is your chance of dying
today always greater than the chance of dying tomorrow. Is there
something that I'm not quite understanding?

OK, this is statistics so I'm probably not explaining it very well.
Assume that the probability that you die is 1%. That is, on any day,
given that you're alive at 00:00, the chance that you will still be alive
at 23:59 is 99%.

It is now 00:00 on 1st January. The probability you will die today is 1%,
as defined. If you make it to the end of the day, the probability you
will die on 2nd January is also 1%. So right now, when we don't know if
you'll survive today (the 1st) or not, the probability you will die on
2nd January is 0.99% - the probability that you will survive 1st January
(99%) times the probability that you will not survive the 2nd (1%).
Similarly the probability you will die on 3rd January is 0.9801%, that
you will die on 4th January is 0.9703% and so on.

So, if asked to bet on which day you will die, given the above I'm going
to put my money on "today". I'm probably wrong, but it's a better chance
than any other day.

You allude to neonatal mortality in your paragraph that refers to the
"zeroth birthday". Very true. But assuming you survive this "boundary
effect", won't the chance of dying stabilise to more or less the same
chance on every date, maybe with a gradual decreasing (the theory you
mention) or a gradual increasing (for elderly people) probability as
each day passes. I don't see what is special about exactly n calendar
years from your date of birth which makes the probability of death
increase on that date and decrease again after it.

That's the point - it doesn't. Assuming a newborn baby, born at 00:00 on
4th March 2021. They are more likely to die on 19th September 2050 than
on 4th March 2051. But, if I choose any arbitrary date, they are more
likely to die on that date than any of the 364 following ones. Nothing
special happens on 4th March 2022, except that I now start counting a new
year. In that year, the day they are most likely to die is the first day
of that year, which is their birthday. In every year, years beginning on
4th March, the day they are most likely to die is the first day of that
period. Add up the years, and the day of the year they're most likely to
die is 4th March.

Now, you will point out that my choice of 4th March (their birthday) to
start the year is arbitrary. I could say that in any year, starting on
21st January, the day they're most likely to die is the first day of that
year. And so if you add it up, the day they're most likely to die is 21st
January. Which is true, but... you've got a part year left over. In they
year starting 21 January 2021, the day they're most likely to die is
*not* the first day of that period, but 4th March - they can't die until
they've been born [which is another mathematical assumption that doesn't
reflect the real world]. So the extrapolation doesn't work - I add up all
the years but have an odd year left over that doesn't fit the rule, and
so can't generalize about "all years". The only way to not have an odd
part-year (where the "monotonically decreasing" rule doesn't apply) is to
choose the day of their birth as the point to start counting.
[Of course, you could point out that I still need to add in the full
years before they're born. Which is true, but that's a full year of
zeroes on every day of the year, and makes no difference to the total].

Also, in your "rather surprising statement", is that increased
probability of dying today rather than tomorrow masked by factors such a
seasonal variation in death date?

Yes. Like most mathematical problems, it is massively over-simplified.
The death rate is far from constant as it varies with age, seasons, other
conditions like wars or pandemics etc. And I'm sure this variation
massively outweighs any bias towards birthday deaths. Which is why it's
dangerous to apply simple mathematical models to the real world - there
are almost always several factors you haven't considered.

And would you expect a 10-year-old to find any of this "blindingly
obvious" to offer it as an explanation? Or anyone except a statistician
to know much about it? I *think* the guy that proposed the question was
a geographer, but I could be wrong.

No. It's far from "blindingly obvious", and I'm far from sure that this
is what the questioner was getting at. It's a rather abstract probability
model resting on some very shaky assumptions, and unless asked in the
context of a maths class I would think it's not the obvious approach to
take. I don't think you need to be a statistician, probability is taught
in some detail in school maths classes, but it's at least GCSE-level if
not A-level.