Some of the greatest breakthroughs in math and science have come when people question the rules. For example, negative numbers questioned the premise that there could not be quantities less than zero. To us in modern times, negative numbers seem intuitive, but to ancient thinkers it was a very difficult idea to contemplate. In fact, the existence of zero itself was very controversial for ages.

There are plenty of other examples. One of the greatest breakthroughs in geometry came when the foundations of Euclidian geometry were questioned. For millennia, mathematicians worked with the 2-dimensional Cartesian plane (x, y) and then 3-dimensional spaces (x, y, z). But it wasn’t until well into the 19th Century that anyone wondered what 4- or n-dimensional math would be like.

Euclid taught us that the interior angles of a triangle always add up to 180 degrees. But when we plot a triangle on, say a globe of the Earth, the triangle appears to bulge slightly and the angles add up to greater than 180 degrees. On surfaces with negative curvature, the angles sum to less than 180 degrees. Questioning that single assumption gave birth to new fields of science that help us understand our universe.

Remember “imaginary numbers,” like 4i or -3i? We were taught that you can never take a square root of a negative number. Then one day in algebra class, they told us “but if you do, just throw an i after the number and keep going.” At some point, someone must have asked, what if you could take the square root of an negative number? What would math be like then? And so a whole new field of mathematics opened up. Imaginary numbers are an essential concept in applications such as systems engineering and quantum theory.

What does this have to do with the Lions and Giants? Recently, I was trying to explain why the effect of home field advantage (HFA) is stronger for closely matched teams and weaker for mis-matched teams. Let’s say for closely matched teams, who would each have a .50/.50 shot at winning at a neutral site, HFA makes the game a .60/.40 proposition. We could describe the strength of HFA as +.10/-.10.

But now take a game where the Lions are playing the Giants. At a neutral site, the game might be something like a .95/.05 proposition in favor of New York. In other words, the Lions would pull of an upset in 1 out of 20 games. But if the game were at the Meadowlands and we apply the same +.10/-.10 adjustment for HFA, the probability the Giants would win would be 1.05, and the probability the Lions would win would be -0.05.

But because probabilities can never be greater than 1 or less than 0, this obviously can’t be the case. Therefore, the effect of HFA must diminish for mis-matched teams. (But if anyone could have a negative probability of winning, it might be the Lions this year!)

Then I thought, let’s question the assumption. Why can’t probabilities be greater than 1 or less than zero? As with negative numbers, or non-Euclidian triangles, or imaginary numbers, let’s just throw away the assumption and keep chugging. What would a universe be like with “imaginary” probabilities?

It’s almost impossible to wrap your brain around such a concept. I don’t know what it would mean. Metaphysical do-overs? Branching timelines? Can the future affect the past?

A quick Google search for “negative probabilities” turns up a number of results, and clearly this has been thought of before. It appears to be an alternative way to explain quantum mechanics. But it was just a weird, stray thought that I thought I’d share.

Right now I've got the Lions with a 10% chance they'll win this weekend, and a 9% chance they'll win next week. This equates to an 18% chance they'll win at least 1 of those 2 games, and an 82% chance they'll finish 0-16.

## Giants, Lions, Home Field Advantage, and Time Travel

By
Brian Burke

published on 12/19/2008
in
home field advantage,
research

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I thought this might be interesting to some people: negative probabilities pop up in quantum mechanics when you're trying to analyze things which can't be measured. For instance, I think a lot of people (well, for some definition of "a lot") have at least heard of the uncertainty principle which says that it's impossible to measure the exact position and velocity of a particle at the same time. There's a similar one for angular momentum, which says that if a particle has a "spin", like an electron, it's impossible to measure the exact direction of the spin axis. You can only compare it to one particular orientation and get an answer of parallel or antiparallel (up or down). If you did try to find the probability that the particle's spin axis points in an exact direction, you might get a negative probability. Physicists justify this by saying that the negative probability doesn't correspond to something you could actually measure, and as long as whenever you calculate the probability of something that is measurable (say, by adding up a bunch of unmeasurable probabilities), you get a positive answer, it's OK. So sure, it's a wacky concept, but nothing to do with time being turned backwards. (Sorry to disappoint ;-)

Hi Brian,aren't you getting into negative probability territory because you're adding/subtracting the same +.10/-.10 HFA correction regardless of the initial neutral field probabilities.

A HFA of three points for two evenly matched sides give the home side around a 59% chance of success,but a team that's a TD superior to another on neutral turf,sees it's win probability only change from 73% on neutral turf to 78% when they host and become 10 point favs.

M

Mark-Right. It's non-linear. You need to add the natural exponent of the change in the odds ratio, not a linear addition of .10, or whatever the advantage is for a neutral team.

I was just using the example as a narrative of what made my mind wander.

By the way, I still think the Lions are so bad that they are reaching into the past and stealing wins from previous Detroit teams. Maybe the 2000 Lions won the Super Bowl and now we've woken up to a reality where the present team has depleted past wins to the point where they only went 9-7 and missed the playoffs.

Prove me wrong!

Or maybe the 2012 Lions are winning the SuperBowl but had to steal a few wins from the 2008 Lions to do it