**Pop Philosophy and Tax Withholding**
*"I know you're basically giving the government a no interest loan when you overpay during the year, but frankly, if I had that money all year I'd have blown it on coffee at Starbucks or something. This way I pay off debt and get a TV."*
--

Matt Boggie
The simplest way to approach the situation is one's instinctive joy at getting a refund or pain at owing more money.

The slightly more sophisticated approach is to point out that for the same tax burden, you'd rather owe money at the end than get a refund, since the latter amounts to a no-interest lone.

The still more sophisticated approach is to point out that,

*from a taxpayer's January perspective*, the fixed variable isn't the tax burden but rather the withholding. If Joe Blow has had $20,000 withheld, then that's still the amount he's had withheld whether his actual burden was $15,000 or $20,000 or $25,000. He'd much rather get $5,000 back (meaning his actual burden was $15,000) than have to pay $5,000 more.

This probably goes in the

*Department of Duh* but I imagine a lot of people don't quite make it from one step to the next.

Along those lines I once seriously blew a job interview because I remained dead convinced that the conventional wisdom on the Monty Hall problem was right and that the people with the unintuitive right answer were the ones who had messed up. When the interviewer gave me the question (the only thing remotely approaching a brainteaser; it was a law firm), I immediately recognized "Monty Hall problem" but I hadn't thought about it in awhile. I remembered concluding that there was something wrong with the unintuitive answer but suddenly, in the interview, I couldn't point it out -- because the unintuitive answer is right after all.

The basic highlights:

1. No matter which of N doors you choose, the prize has a 1/N chance of being behind that door regardless of the host's future behavior,

*assuming (big assumption!) that the prize distribution is completely random*
2. If the premise is that the host always chooses (arbitrarily) N-2 doors to reveal after your initial selection, then the probability that the door neither you nor he choose is the correct one does indeed go up.

2a.

*But*, this only works if the N-2 doors can be (from your perspective) any of the N-1. Important distinction for the case where N=10,000 and you run the game 10,000 times: Is the remaining choice just between door #1 (which you picked) and some other of the doors, varying with each trial? Or does the remaining choice always

*happen* to be between door #1 and door #10,000? In the second case the prize distribution wasn't random after all.

(Or, as close as I could come to defending the conventional wisdom in the interview: instead the host always choosing to reveal door #2 or door #3, it may be that the host chooses to reveal door #2 if and only if door #2 is actually empty, and happens never to choose to reveal door #3. If

*that* were the case then you really would have a 50/50 chance.)