If you read the previous article on this topic, then I imagine you were quite piqued by the nature of its contents. How we use mathematics to find a mall parking spot is not a typical thing you would hear people discussing at their Christmas parties. Yet I think anyone with a modicum of human interest would find this a most curious topic of conversation. The reaction I usually get is one of “Wow. How do you do that?”, or “You can really use mathematics to find a parking spot?”
As I mentioned in the first article, I was never content to get my degrees in mathematics and then not do anything with them other than to leverage job opportunities. I wanted to know that this newly found power that I studied feverishly to obtain could actually inure to my personal benefit: that I would be able to be an effective problem solver, and not just for those highly technical problems but also for more mundane ones such as the case at hand. Consequently, I am constantly probing, thinking, and searching for ways of solving everyday problems, or using mathematics to help optimize or streamline an otherwise mundane task. This is exactly how I stumbled upon the solution to the Mall Parking Spot Problem.
Essentially the solution to this question arises from two complementary mathematical disciplines: Probability and Statistics. Generally, one refers to these branches of mathematics as complementary because they are closely related and one needs to study and understand probability theory before one can endeavor to tackle statistical theory. These two disciplines aid in the solution to this problem.
Now I am going to give you the method (with some reasoning–fear not, as I will not go into laborious mathematical theory) on how to go about finding a parking spot. Try this out and I am sure you will be amazed (Just remember to drop me a line about how cool this is). Okay, to the method. Understand that we are talking about finding a spot during peak hours when parking is hard to come by–obviously there would be no need for a method under different circumstances. This is especially true during the Christmas season (which actually is the time of the writing of this article–how apropos).
Ready to try this? Let’s go. Next time you go to the mall, pick an area to wait that permits you to see a total of at least twenty cars in front of you on either side. The reason for the number twenty will be explained later. Now take three hours (180 minutes) and divide it by the number of cars, which in this example is 180/20 or 9 minutes. Take a look at the clock and observe the time. Within a nine minute interval from the time you look at the clock–often quite sooner–one of those twenty or so spots will open up. Mathematics pretty much guarantees this. Whenever I test this out and especially when I demonstrate this to someone, I am always amused at the success of the method. While others are feverishly circling the lot, you sit there patiently watching. You pick your territory and just wait, knowing that within a few minutes the prize is won. How smug!
So what guarantees that you will get one of those spots in the allotted time. Here is where we start to use a little statistical theory. There is a well-known theory in Statistics called the Central Limit Theory. What this theory essentially says is that in the long run, many things in life can be predicted by a normal curve. This, you might remember, is the bell-shaped curve, with the two tails extending out in either direction. This is the most famous statistical curve. For those of you who are wondering, a statistical curve is a chart off of which we can read information. Such a chart allows us to make educated guesses or predictions about populations, in this case the population of parked cars at the local mall.
Charts like normal curve tell us where we stand in height, let us say, with respect to the rest of the country. If we are in the 90th percentile in regard to height, then we know that we are taller than 90% of the population. The Central Limit Theorem tells us that eventually all heights, all weights, all intelligence quotients of a population eventually smooth out to follow a normal curve pattern. Now what does “eventually” mean. This means that we need a certain size population of things for this theorem to be applicable. The number that works very well is twenty-five, but for our case at hand, twenty will generally be sufficient. If you can get twenty-five cars or more in front of you, the better the method works.
Once we have made some basic assumptions about the parked cars, statistics can be applied and we can start to make predictions about when parking spots might become available. We cannot predict which one of the twenty cars will leave first but we can predict that one of them will leave within a certain time period. This process is similar to the one used by a life insurance company when it is able to predict how many people of a certain age will die in the following year, but not which ones will die. To make such predictions, the company relies on so-called mortality tables, and these are based on probability and statistical theory. In our particular problem, we assume that within three hours all twenty of the cars will have turned over and be replaced by another twenty cars. To arrive at this conclusion, we have used some basic assumptions about two parameters of the Normal Distribution, the mean and standard deviation. For the purposes of this article I will not go into the details regarding these parameters; the main goal is to show that this method will work very nicely and can be tested next time out.
To sum up, pick your spot in front of at least twenty cars. Divide 180 minutes by the number of cars–in this case 20–to get 9 minutes (Note: for twenty-five cars, the time interval will be 7.2 minutes or 7 minutes and 12 seconds, if you really want to get precise). Once you have established your time interval, you can check your watch and be sure that a spot will become available in at most 9 minutes, or whatever interval you calculated depending on the number of cars you are working with; and that because of the nature of the Normal curve, a spot will often become available sooner than the maximum allotted time. Try this out and you will be amazed. At the very least you will score points with friends and family for your intuitive nature.
Source by Joe Pagano
