The function that approximates the best ideal solution can be approximated with the log function. It's kind of like why the taylor polynomials means that e^pi*i = -1

I don't think that was exactly a lay explanation. Was interested myself.

Yeah, it totally wasn't. I failed.

Okay, so one of the really interesting properties of mathematics is that complicated functions can sometimes be simplified by simple functions.

For example, the gamma function is an extension of the factorial function (4! = 432*1).

http://upload.wikimedia.org/math/a/c/5/ac57cb1b5db9b61155d86...

Amazingly, you are able to represent one idea in another, different way.

So, with the secretary problem, you are able to break down the optimal solution as this:

http://upload.wikimedia.org/math/6/f/4/6f4d3a757d9efe51b8b17...

The | notation means "given". So x|b means "x, given b".

For these abstract formulations, they are reduced to already-existing calculations..

Finally, you get the big sigma symbol. This is called a Summation. Summations tend to be able to be approximated using integrals, since an integral is the area under the curve, which is also a summation of sorts.

Many integrals can be defined using the "log" function, which has base "e". In this case, when you add in "infinity" for the summation, it converges towards 1/e.

Hope this helps some. If it doesn't, well, I was fired from my job TA'ing calculus =B so..