"The only poll that matters is on election day," goes the old cliché. It's a reassuring sentiment for Mitt Romney supporters, given that recent polling shows their candidate running behind in many key swing states. And there's a kernel of truth to it: There's absolutely a chance the polls are all wrong.

But just how wrong would current polls have to be for Romney to win the election? Robert Erikson, a prominent forecasting specialist at Columbia, and his colleague Karl Sigman did the math. They estimate that if the current polls are correct, Obama has a 99.9 percent chance of winning.

That's impressive, but not that remarkable given that Obama has more electoral votes from safe states than Romney does. That means that if you flipped a coin to determine who won different battleground states, Obama would still have an 83.9 percent chance of winning. The geography is just more favorable to him.

But what happens if the polls are wrong, and biased (in the statistical sense, not the ideological sense) in Obama's favor? If they're two points too favorable to Obama, Erikson and Sigman estimate that there's still an 65.9 percent chance Obama wins; if they're four points off, there's a 1.7 percent chance he wins, making a Romney win a near certainty. The following table summarizes these results:

Source: Erikson and Sigman

A four-point swing in the final days isn't unthinkable. Erikson and Sigman note that a swing nearly that big happened in the national vote in 2000, with almost 4 percent of voters swinging toward Al Gore following the discovery of George W. Bush's DUI arrest.

The problem was that the swing bypassed battleground states, where the candidates had already spent more time campaigning, indicating that those voters' opinions were more fixed in the last weeks than those of voters in safe states. So a four-point swing in battleground states is a little harder to imagine than a swing that large in the national vote.

"Unheard of" doesn't mean "impossible," of course, but it does suggest that Romney has tough odds to overcome Tuesday.