The binomial theorem and herd immunity

Godfrey Hardy once called out another statistician's faulty analysis of Mendelian genetics, writing:

"A little mathematics of the multiplication table type is enough..."

I wish to call out some fatally flawed amateur assumptions about herd immunity. Please stay with me, because this has lockdown relevance. Be warned that the binomial theorem seems counter-intuitive, but it's a corner-stone of mathematics.

At one time, I was between jobs, and I worked as a casual (supply) teacher. I was amused once to hear some of my charges discussing me on a bus. One claimed I was their science teacher, another knew me as a computing teacher, and another thought I was their maths teacher. The fourth said "Naaah, he's Angus' dad."

They were all correct, but at the school where I taught maths, my Year 9 kids quickly worked out that I gave intellectually honest answers. I also had them to the point where a number of them came up to me in the playground to ask about maths problems I had floated past them. That's the payoff for being intellectually honest.

One of those kids, a boy, later asked me in class "what's the use of the binomial theorem?" and I said I would look into it. The next day, I gave an adequate answer. Now here's a Good Answer:

(a + b)^2 = a^2 + b^2 + 2ab. That's one form of the B.T. I keep hearing GladnBradnScotty bumbling about 70% vaccination being the Gold Standard, but consider a hundred random meetings after that time. Here are the odds: I will call single vax and no vax Nvx and double vax Dvx. P (Dvx x Dvx) = 0.7 x 0.7, so 49% might be OK. That may not be so, but let's pretend it is.

P (Nvx x NVx) = 0.3 x 0.3, so 9% are in a handbasket together. The balance of cases, 42%, involve a contact between a vaxed and a non-vaxed person (2 x 0.3 x 0.7, if you prefer). I call those people at risk, and notice that 42% > 30%. The 49% are OK, the 9% are self-selected and gone, but the 42% bloc is where the mayhem will occur. The vaxed can carry and spread the virus to the 21% of unvaxed that they encounter (and some of the vaxed can still be infected. Putting it another way, the 30% of unvaxed have a 70% chance of encountering a vaxed person: that's 21% another way.

Even with 80% vaxed, that means 16% vulnerable, and 90% vaxed leaves 9% vulnerable.

Now one of the three pollies named above once worked for me, and said numpty's papers were marked by me "not to be reemployed" due to incompetence, and I have only done that to one subordinate, ever. The other two are, in my estimation, far worse. These people should only be allowed to put the bins out, and even then, only if they are supervised.

There's a storm coming, and I want to be able to point back to this and say "Told you so!" There is no way in a sane world that you let decision-makers choose options when they haven't run the numbers. The Binomial Theorem does not make mistakes.

Please fasten your seatbelts...