This program takes

• $g$ in $]0, infty[$, a random time gone since something began.

• $p$ in $]0, 1[$, the portion of time that you consider to have elapsed.

and produces the interval in which that something should end.

The method is taken from

William Poundstone; The Doomsday Calculation; Little, Brown Spark, 2019

The relevant part of this book can be read as

Amazon preview https://www.amazon.com/Doomsday-Calculation-Equation-Transforming-Everything-ebook/dp/B07J4WCSMR

If you prefer an audio introduction, watch on YouTube

The Doomsday Calculation: Book Trailer https://youtu.be/jr693Q6M8OY

Also, an interview with the author is available as Michael Shermer with William Poundstone — The Doomsday Calculation (SCIENCE SALON # 76) https://youtu.be/M0tHz4BdrvA

In pp.14-18 of Poundstone's book there is a description of a prediction J. Richard Gott III made regarding the fall of the Berlin wall.

The input data are

• In 1961 the wall was built.

• In 1969 Gott visited the wall, which he considers that

to have been a random moment in time.
• He proposes to make a prediction with a 50% level of confidence,

which is equivalent to saying that at least 25% of the total duration
is assumed to have elapsed.

so that

• `gone` is 1969 - 1961 = 7 years.

• `portion` of the time gone is a half the 50% or 1/4.

which produces the interval $[3, 24]$, which translates to

• There is a 50% chance that the wall falls sometime between

$1969 + 3 = 1972$ and $1969 + 24 = 1993$ .

The wall actually fell in 1989.

Since the method is controversial, read the book before applying
it to your personal problems.
Pay attention to the randomness of `gone` and time scale invariance.

Consider something of interest that began at time `start`.
Take the origin 0 of the time axis to be a random time `now`.
Assuming that the end does come, it has to be within finite time from `now`.
Let that maximum time be the unit $1$.
Since the maximum time is unknown,
the time unit 1 is unknown in any time scale.

Consider the $100 , q$% confidence interval within which the period ends.

Let $p := (1 - q)/2$.
The *end` falls somewhere within the confidence interval $[p, 1-p]$
with probability $q$.
`

`
`

`
The `end` will come with probability 1 sometime after 0,
but with probability $q$ only after $p$ and before $1-p$.
This means that with probability $q$ the time elapsed between `start` and
`now`
equals $p$.
`

`
Let the time `gone` be $g :=$ `now` $-$ `start`.
This quantity comes with a time scale since `now` and `start` are measured in
some unit such as days or years.
Now let $ell$ be the time corresponding to $p$ in the scale of $g$.
Then $ell/g = p/(1-p)$ so that
$$
ell = frac{p}{1-p} , g
$$
`

`
Similarly, the time corresponding to $1-p$ in the scale of $g$ is
$$
u = frac{1-p}{p} , g
$$
`

`
Now the end will come neither until the next moment of *now`, which is 0,
nor after the earliest end within the confidence interval which is at $p$.

To sum up, with $100 , q$% confidence $l := g , r$ and $u := g / r$, with $r := p/(1-p)$.

Implementing this method, I make a function called *gott` which takes
`start`, `now`, `gone` and returns `l` and `u`, where
`

`
• `l`: shortest time from `now` that the event may take place
• `u

`
both with $100 , q$% confidence.
`

`
Helpers.
`

`
`

`
Here is the program.
`

`
`

`
`

This problem has been taken from Poundstone's book. In pp.14-18 there is a description of a prediction J. Richard Gott III made regarding the fall of the Berlin wall.

The input items are:

`
• .1961. *start` : floating time when it began

• .1969. `now` : floating present time

• .0.25 `portion` : floating portion `gone`, between 0 and 1

It is crucial that `now` may be considered a random moment
in time line after `start`.
This is a special case of the Copernican principle.

1961 is when the wall was built. 1969 is the time in which Gott visited the wall. He considers this point to be a random time in the period of existence of the wall. 0.5 is the confidence level; the prediction is made so that there is a 50% chance that the end of the wall will fall within the time interval to be produced below.

gives

f 1971.666667 1993.000000

meaning that there is a 50% chance that the wall will fall sometime between 1972 and 1993.

"His 1967 prediction was that there was a 50 percent chance that the wall would stand at least 2.67 years after his visit but no more than 24 years." (p.16)

To recalculate with a confidence level of 95% as in p.19 of the book,

which gives

f 1969.205128 2281.000000

or between 1969 and 2281; reasonable but uninteresting.

Now try the relationship duration between Diana and Charles that comes as the first example in the book, p.3.

giving

f 1994.268519

or after 1994.
`future.hi` is irrelevant, considering the couple's life span.
Note that the value of `Gone` is set to 0.1 rather than 0.05,
even though the level of confidence is stated to be 90%.

Regarding this result the book says

"Gott's formula predicted a 90 percent chance that the royal marriage would end in as little as 1.3 more years."

which is misleading: if a 90% confidence level is assumed the split will not take place at least until 1.3 years from "now."

"The split was formalized on August 28, 1996."

Poundstone book p.82.

• 1934-09 The Third Reich proclaimed.

• 20 months before, Hitler rised into power.

• 95% confidence.

f 1934.792735 1999.750000

"A Copernican would have predicted the Nazi state to survive somewhere between another two weeks and another sixty-five years (at 95 percent conficence). The Third Reich lasted another eleven years."

Dates are in YYMMDD.

190726 Brasilia Time (BRT)
` Project proposal from a software company FS to `
` a chemical company UL, both in Brazil, upon a request from UL.`
190815 Received a message from FS that UL has
` not yet made a decision.`
` 19 days have passed since the proposal.`

f 4.750000 76.000000

So there is an 80% chance that UL will not give its decision within 5 days.

190822 No notice yet, as predicted.

Since 76 days are about 2.5 months, the chance that a notification will arrive by the end of November is 80%.

191112 Project cancelled due to a change in market prices.

This is also as predicted.

190722 HE, a Japanese company, asked me to comment on
` its employee's work, so they can make a decision on her.`
190724 HE sent me related data.
190730 I turned in my report to HE.
190801 HE asked me to wait for their decision.
190816 Today.

f 20.250000

So there is an 80% chance that HE will not send me their decision before 190820.

190822 Received HE's decision on its employe.

This is as predicted.

f 2022.894737 2027.222222 2037.500000 2315.000000

There are 90, 80, and 70% chances that Japan will not go into a war until 2023, 2027, and 2038, respectively.