Want to be an actuary? Odds are, you’ll fail the test

Note that the formulas in question are the opposite of interesting formulas. They're just made-up regulatory rules about how much capital an insurance company has to hold in order to sell insurance. Everyone agrees that some amount of capital needs to be held. Everyone, except maybe some of the "actuarial scientists" themselves, understands that this isn't a scientific endeavor like discovering the laws of General Relativity, it is an arbitrary and convoluted "rule of thumb" type of formula, but it's better than nothing because you really do need to force insurance companies to hold some minimum capital and you need some kind of standard formula to calculate that capital in order for there to be a level playing field between insurance companies.

I think actuarial science undergrad students choose that career because it is one of the safest choices for someone who is good at math but has no interest in science or technology. They are allured by the promise of steady employment and a $100,000 salary, and vague visions of being a high paid "math AND business expert" for a big insurance company where they will get to make decisions involving large sums of money with scientific precision using advanced mathematical concepts.

By the time your young actuary has started working, they are already too deep in to be able to change careers. They made the "safe" choice, now they have to live with it. At first everything seems new and exciting. But 10 years into it, they have settled down roots and family somewhere in the middle of Iowa (because that's where the insurance company is located), knee deep into spreadsheets calculating "Solvency II" formulas for "quarter end." They don't get to make any business decisions, they barely understand how their employers' business even really works, and their career has plateaued at a mediocre level despite having spent 10 years writing all of the available actuarial exams. Unfortunately, that is the only employer of actuaries in town, so they are completely marooned. They spend their free time learning the latest tips and tricks about Excel VBA programming, watching the movie "About Schmidt" repeatedly, and mistakenly envying their peers who work in banking instead of insurance

let pc = purchased_collision, pd = purchased_disability

we have:

2 * P(pc) = P(pd) P(pc & pd) = 0.15

so

P(pc) * P(pd) = 0.15 2 P(pc)^2 = 0.15 --> P(pc) = 0.274, P(pd) = 0.548

So the probability of neither pc nor pd is

!P(pc | pd) = 1 - P(pc) - P(pd) + P(pc & pd) = 0.328 ~= 0.33?

The addition of P(pc & pd) was to take account of the double counting of pc and pd.

Please let me know if I've made a mistake!

Two observations:

1. The test was ridiculously easy. Junior high–level math and reading. And yet teacher candidates still failed it.

2. For K-12 teaching, the knowledge that was tested was pretty much irrelevant for the job. What matters for more than being able to find the length of the hypotenuse of a right triangle or to comprehend an expository paragraph (which is not to say that these aren't useful skills) is the ability to manage a classroom. Almost none of the teacher prep I had was geared towards that single most important skill (which I, unfortunately, was not so great at) and which is what will make the difference between succeeding and failing in a K-12 teaching career.

I think this generalizes to other fields as well. Think about all the tree-traversal BS we've all gone through in a software engineer interview and how little that relates to the actual work that we do and how the hard stuff is much less the programming and more the learning the code base and business.