A few years ago I calculated odds of the end of the world, but I didn’t show my work. It helps to have math notation, which I can get on Discourse.
The rule of three
John D. Cook showed how the odds of something not happening can be estimated from the number of times something hasn’t happened already. So if you’ve read n pages without a typo, you can estimate (with a 95% confidence interval) that there won’t be a typo on the next page you read is 3/n. So the odds that there will be a global nuclear war next year can be estimated by the number of years since nuclear weapons were first developed:
p = \frac{3}{(2025-1945)} = \frac{3}{80} = 0.0375
You might argue that the start should be a few years after 1945 since only one of the super powers had a nuclear arsenal. So maybe start with 1949 when the Soviet Union began building their capability which comes to ~4%.
Median time to failure
How many times can we roll the dice before catastrophe? Graham Kemp on Mathematics—Stack Exchange gives the formula for median number of rolls (n) before rolling a particular number (such as 6):
n = \frac{-1}{\log_2 (5/6)}+1
Generalizing for any probability of failure (p), the median number of tests before failure (m) is:
m = \frac{-1}{\log_2 (1-p)}+1
Calculating base 2 log with natural log
I use Numi to do calculations and it only has a natural logarithm function (ln) and \log_{10} (log). Fortunately, it is possible to convert from one base to another. Elizabeth Stapel gives the formula on a Purplemath post:
log_b(x)= \frac{log_d(b)}{log_d(x)}
So we can reorganize our median time to failure function to use natural log:
m = \frac{-\ln(2)}{\ln(1-p)}+1
Putting it all together
Finally we can add in the rule of three calculation for estimating probability:
m = \frac{-1}{\log_2 (1-(3/n))}+1
Using Numi’s notation, that’s:
-(ln(2))/ln(1-(3/(2025 - 1945))) + 1
On the command line:
$ numi-cli -- '-(ln(2))/ln(1-(3/(2025 - 1945))) + 1'
19.14
Bonus: minutes to midnight
To match the presentation of the Doomsday Clock, I used the proto-city of Jericho as the start of civilization, which was founded sometime in the 9th millennium BC. So we can scale our clock thusly:
This, of course, is unhelpful precision. Since we are making a (very rough) estimate, we can say each minute is 7 years, which would represent a somewhat younger civilization (10,000 years or so). If my estimate of the median time to destruction (19 years) holds water, I’d set the Doomsday Clock to 3 minutes to midnight:
