# Finding a parking space in a shopping center using mathematics – Part II

If you read the previous article on this topic, I imagine you were quite annoyed by the nature of its content. The way we use math to find a parking spot in a mall is not something you’d typically hear people talk about at their Christmas parties. However, I think anyone with a modicum of human interest would find this a very curious topic of conversation. The reaction I usually get is one of “Wow. How do you do that?” or “Can you really use math to find a parking space?”

As I mentioned in the first article, I was never satisfied with getting my math degrees and then doing nothing with it other than taking advantage of job opportunities. I wanted to know that this newfound power that I feverishly studied to obtain could be of personal benefit to me: that it could be an effective problem solver, and not only for highly technical problems but also for more mundane ones such as the case at hand. Consequently, I am constantly inquiring, thinking, and looking for ways to solve everyday problems, or use math to help optimize or expedite an otherwise mundane task. This is exactly how I found the solution to the mall parking spot problem.

Essentially, the solution to this question arises from two complementary mathematical disciplines: Probability and Statistics. In general, one refers to these branches of mathematics as complementary because they are closely related and probability theory needs to be studied and understood before statistical theory can be approached. These two disciplines help in solving this problem.

Now I’m going to give you the method (with some reasoning, fear not as I won’t get into some laborious mathematical theory) on how to find a parking spot. Give it a try and I’m sure you’ll be amazed (just remember to drop me a line about how cool this is). Ok, to the method. Please understand that we are talking about finding a spot during peak hours when parking is hard to find; obviously, there would be no need for a method in different circumstances. This is especially true during the holiday season (which is actually the time of writing this article, how fitting).

Ready to try this? Come on. The next time you go to the mall, choose a waiting area that allows you to see a total of at least twenty cars in front of you on each 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. Look at the clock and note the time. Within a nine minute interval from the time you look at the clock, often well before then, one of those twenty or more places will open. The math pretty much guarantees it. Every time I try this and especially when I demonstrate it to someone, I am always amused by the success of the method. While others feverishly circle the lot, you sit there patiently watching. You choose your territory and simply wait, knowing that in a few minutes the prize is won. What a smug!

So what guarantees you’ll get one of those spots in the allotted time? This is where we start to use a bit of statistical theory. There is a well known theory in Statistics called 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. You may remember that this is the bell curve, with the two tails extending in either direction. This is the most famous statistical curve. For those of you wondering, a statistical curve is a graph from which we can read information. Such a graph allows us to make educated guesses or predictions about populations, in this case, the population of cars parked at the local mall.

Graphs like the normal curve tell us where we are in elevation, say, relative to the rest of the country. If we are in the 90th percentile for 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 IQs in a population eventually smooth out to follow a normal curve pattern. Now, what does “eventually” mean? This means that we need a population of things of a certain size for this theorem to apply. The number that works very well is twenty-five, but for the case at hand, generally twenty will be enough. If you can have twenty-five cars or more in front of you, the better the method works.

Once we’ve made some basic assumptions about parked cars, statistics can be applied and we can start making predictions about when parking spots will become available. We cannot predict which of the twenty cars will come out first, but we can predict that one of them will do so within a given period of time. This process is similar to what a life insurance company uses when it is able to predict how many people of a certain age will die in the following year, but not which ones will. 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 cars will have overturned and will be replaced by another twenty cars. To reach this conclusion, we have used some basic assumptions about two parameters of the Normal Distribution, the mean and the standard deviation. For the purposes of this article I will not go into detail about these parameters; the main goal is to show that this method will work very well and can be tried next time.

In short, 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 information). ). Once you’ve set your time slot, you can check your watch and make sure a spot will be available in a maximum of 9 minutes, or whatever interval you’ve calculated based on how many cars you’re working with; and that due to the nature of the Normal curve, a spot will often become available before the maximum allotted time. Try this and you will be surprised. At the very least, you’ll score points with friends and family for your intuitive nature.