John D. Cook

You want to consider, based on a distribution of potential fly populations, what the odds are that sampling two of them will fail to capture two specific ones. For example, you can state with certainty that the population is above 3.

Closely related (though not quite the identical problem): https://www.johndcook.com/blog/2010/03/30/statistical-rule-of-three/

Applying that directly, we would estimate that the odds of a fly in your house being marked are less than 150%, which tells us that you should be taking larger samples.

(With only two marked flies, this estimate will always be less informative than the fact that you sampled n flies -- it will tell you that the population is probably greater than 2/3 of n, while the sampling procedure tells you that the population is necessarily at least n+2. But as you mark more flies, the estimate will be more informative.)

What software are you using that once or twice a year doesn't count as "rarely"?

My Viaplay app stops streaming and tells me I'm not connected to the internet several times a day (when I'm obviously still connected to the internet.)

Several times a week my phone's Netflix state doesn't match my TV's state, so I'm either not presented with the controls or when I press them they do nothing.

Cosmic bit-flips are rare enough to be newsworthy.
https://www.johndcook.com/blog/2019/05/20/cosmic-rays-flipping-bits/

If the faults were with the hardware, I'd expect the errors to be spread around the software that I use. But Firefox and Gmail always work for me (even if the JavaScript I'm served makes the machine unresponsive), whereas Netflix, Viaplay and HBO constantly misbehave.

To the section "Popular Math Books" I would add almost anything by Julian Havil. John D. Cook referred to him as a writer of "serious recreational mathematics" [1]. I would probably put his "Nonplussed!" and "Impossible?" books in the "Level: Easy" group, with the others at least in "Level: Medium". "Gamma" is one of my favorite serious recreational math books.

If you like getting into the nitty-gritty of problem solving then check out the books of Paul Nahin. They vary between "Level: Medium" and "Level: Difficult", with many of them reveling in the solution of equations, and integrals in particular. Although he recognizes the need for proofs, he makes a point of avoiding them in his books.

[1] https://www.johndcook.com/blog/2019/09/29/a-sort-of-mathematical-quine/

For some definition of “sufficiently introductory”, I’d recommend starting with the first chapter of John Nolan’s book Stable Distributions [0] (20 pages), which presents the class of distributions to which sums of iid random variables converge and builds up to a version of the generalized CLT.

Note that this generalization of the classical CLT relaxes the requirement of finite mean and variance but still requires that the summed random variables are iid. There are further generalizations to sums of dependent random variables. John D. Cook has a good blog post that gives a quick overview of these generalizations [1].

0. https://edspace.american.edu/jpnolan/wp-content/uploads/sites/1720/2020/09/Chap1.pdf [PDF]

1. https://www.johndcook.com/blog/central_limit_theorems/

I found about posits.

https://www.johndcook.com/blog/2018/04/11/anatomy-of-a-posit-number/

Perhaps CFP8 are parameterized 8-bit posits where the parameter is the value es. The larger es is, the greater the dynamic range is at the expense of precision. Two examples:

posit<8, 0> (es = 0) has as largest positive number 64 and the smallest positive number 1/64.

posit<8, 1> (es = 1) has as largest positive number 4012 and the smallest positive number 1/4012.

The formula for the largest positive number for 8-bit posits is:

2 ^ 2 ^ es ^ 6.

posits don't have NaNs and only one infinity (±∞), so they can use more of the 8 bit values for numbers than floating point numbers.

I wonder: is CFP8 = posit<8, es>?

> Also disappointing that they didn't follow up the incorrect responses with corrections. Like if you told gpt that "sorry, your answer is wrong because the stock market is closed on saturday", it would come up wiht a new answer that takes that into account.

If you have to keep correcting the tool yourself, you won’t arrive at the truth but at the limits of your own knowledge. You’ll have no basis to know which answer is the one you can finally trust.

That mode of operation reminds me of the Gell-Mann amnesia effect.

https://www.johndcook.com/blog/2021/01/18/gell-mann-amnesia/