# Punk Rock Operations Research

## the license plate game: the raw numbers

Posted by Laura McLay on January 20, 2012

My last post discussed how one might estimate how many state license plates one would expect to see on a road trip. I made a spreadsheet to compute the probability of seeing each state license plate.

Assumptions

1. The probability of seeing a state license plate A in another state B depends on the distance between their state capitals. It is scaled by the  number of licensed drivers in state A. (This indirectly means that the probability does not depend on how long we are in a state).
2. Seeing state license plates A, B, etc. are independent from other license plates in a given state D.
3. Seeing given state license plate A is independent when driving across states B, C,…
4. We do not adjust for round trips.

The distance between state capitals was found here. The number of licensed drivers per state is here. I estimated the odds of seeing a license plate from state A in state B is captured by this formula:

P = exp(-K * (Distance from A to B in miles) / # of licensed drivers)

with K = 7000 – 2000*Summer01 – 1000*ExpensiveGas01. Summer01 is 1 if it is summer break and 0 otherwise. ExpensiveGas01 is 1 if it gas is “expensive” and AAA predicts that road trips will be down and 0 otherwise. I didn’t have time to properly identify a meaningful formula or calibrate the parameters. Suggestions here are welcome!

Validation

• We predicted 28.3 states for our summer trip from Richmond to Chicago. We saw ~35. Here, the discrepancy seemed to be the amount of time we spent in each state. We went through fewer states, but was in each state (especially Kentucky and Indiana) a relatively long time.
• We predicted 26.8 license plates for our winter trip from Richmond to Vermont. We saw 26. Not bad!

The results make me conclude that the first assumption is probably not true: the probabilities do depend on how long we are in a state. When driving to Vermont, we went through many (8) little states. When driving to Chicago, we went through fewer (5) states but were in each state for longer.  Moreover, many of the Midwest states are not “destination” states. Take Indiana for instance. I love Hoosiers as much as the next person, but Indiana truly is the “Crossroads of America”–it’s a state that many people from other states drive through. It’s a better place to spot license plates than, say, Delaware. I didn’t take that into account.

Below is a detailed review of our winter trip numbers. It indicates the predicted probability of seeing each state license plate and whether we actually saw it. As asterisk (*) indicates whether the model is “off”–whether we (1) did not see a state with probability greater than 0.5 or (2) did not see a state with a probability of 0.5 or lower.

A copy of my spreadsheet is here if you want to see how I computed the numbers.

 State Cumulative probability of seeing each state States we saw Alabama 0.671 * Alaska 0 Yes * Arizona 0.065 Arkansas 0.060 California 0.961 Yes Colorado 0.083 Yes * Connecticut 1 Yes Delaware 1 Yes District of Columbia 1 Yes Florida 0.999 Yes Georgia 0.971 Yes Hawaii 0 Idaho 0 Illinois 0.973 Yes Indiana 0.950 * Iowa 0.056 Kansas 0.028 Kentucky 0.710 * Louisiana 0.236 Maine 0.565 Yes Maryland 1 Yes Massachusetts 1 Yes Michigan 0.990 * Minnesota 0.269 Mississippi 0.060 Yes * Missouri 0.563 Yes Montana 0 Nebraska 0.001 Nevada 3.53E-06 New Hampshire 0.911 Yes New Jersey 1 Yes New Mexico 3.14E-05 New York 1 Yes North Carolina 0.999 Yes North Dakota 0 Ohio 0.998 Yes Oklahoma 0.032 Yes * Oregon 0.0006 Pennsylvania 0.999 Yes Rhode Island 0.863 * South Carolina 0.878 Yes South Dakota 0 Tennessee 0.841 * Texas 0.983 Yes Utah 5.61E-05 Vermont 1 Yes Virginia 1 Yes Washington 0.037 West Virginia 0.416 Wisconsin 0.671 Yes Wyoming 0

1. ### Laura McLaysaid

My husband is impressed with this model. He writes:

“I like the first order model we came up with, it’s pretty good for what it does! Of course, it would be a better empirical fit if we had a few more datapoints and spent some effort calibrating.

Ultimately what the license plate game does is sample traffic, and I’d guess that traffic volume is the significant factor we’re not capturing. On our VT trip we picked up a dozen license plates in that traffic jam on I-95-495 in D.C., but not very many subsequently in rural PA or NY. I tried googling for average traffic volume data on the web, and I found some that indicated urban areas carry upwards of 250000 cars per day, while some rural interstates in Idaho and NV carry only 2000-6000 per day. So a short 20 minute trip driving around Richmond on 295 would certainly be expected to turn up more plates than a 4 hour drive across the unpopulated western deserts.

My next stab at a better approximation would be a massive weighted graph of interstate nodes and urban areas, with the weights reflecting traffic volume. This model gives me a headache already, and would not fit in a neat blog post like yours!”

2. ### Paul A. Rubinsaid

I think this should really be some sort of non-homogeneous Poisson process (with rate dependent on factors you’ve mentioned, and maybe a few others). Driving skill is also a consideration. You don’t see many MI plates in NY or MA due to very short life expectancy.

3. ### David Blakesaid

TheIndiana result is not surprising but the Michigan result is. We you in the far south of
Indiana?

4. ### Laura McLaysaid

David, We were happy to see such high odds of seeing California, Florida, and Georgia plates, despite not driving near these states. We always see those plates on trips, and the model reflects this. The first order approximation is a good start. I agree–the high Michigan probability is surprising.

5. ### Dr Data's Blogsaid

[...] – How many state license plates do you expect to see in a road trip? Depends on (a) how many miles you drive, and (b) number of licensed drivers per [...]

6. ### Ramonnsaid

Recently as I was driving on a California freeway a car pulled in front of me. I noticed the vanity plate was the same as my last name. What are the odds of this happening?