But really, my quibble is not with the math at all, but with the implicit cognitive dissonance. Voting takes, what, 20 minutes? This represents 0.0038% of the all the minutes in a year. Unless the same rational calculations are applied to the other 525,580 minutes, it seems a bit bizarre to insist that voting reach some mathematical likelihood of significance threshold when clearly many other daily, weekly, monthly, and yearly activities do not. Indeed, I imagine that it took longer for the emailer to bang out all those equations than it would have to vote!
I think the key reason for why most people vote was hit on in the other email I posted:
There is a undeniable biological drive for humans to want to be part of a larger group, whether that be a church, allegiance to a local sports team, whatever. Certainly it made survival sense in our evolutionary past, and the psychological benefits of it are still with us, and voting is just one more way of engaging in society. At the very least, while probably not as good as random-guy-with-baby, random-guy-voting is probably not a bad way to meet women.
Basically, people feel good about themselves when they vote, in the Ben Barber "participatory democracy" sense (even if the election is a huge blowout, there are important individual and group benefits to having people push the lever).
Of course, why most people vote is no argument either for or against the rationality of voting, it is simply my guess at an explanation. But I would submit that rational vs. irrational behavior only makes sense in the context of an end goal. For example, if you have $1000 sitting around and have decided that your goal is to attempt to increase that dollar amount, there are rational decisions that could be made -- paying off your mortgage, putting it in a savings account, investing it in the stock market -- which will all have pros and cons which could be rationally determined. There are also, of course, irrational ways of attempting to grow your money -- giving it to me to bet in Saratoga, for example.
But if we eliminate the goal, or at least define so fuzzily as 'the happiness in your life', the concept of rational and irrational goes out the window. If you have $1000 and are deciding between the goal of investing and the goal of pleasing your wife by getting her a really nice birthday present, there's really not a rational vs. irrational decision to be made, but rather a weighing of priorities. I would argue that voting is much more a 'weighing of the priorities of my free time' than a 'rational calculation of if my vote will matter.'