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.
- 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).
- Seeing state license plates A, B, etc. are independent from other license plates in a given state D.
- Seeing given state license plate A is independent when driving across states B, C,…
- 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!
- 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|
|District of Columbia||1||Yes|