Markov Chains, Matrices, Mutations, Magic, and More!

Hi everyone, Maria and Courtney here! Today was another exciting day of both math and park exploration, and it included further study of probability concepts, project work, and time in Epcot and Hollywood Studios.  Here’s a quick look at all that we learned and did today:    

Markov Chains & Matrices

An example tree diagram that we studied
this morning in class.  The branch values
display the probabilities of each
event in the chain and the values to the
right show conditional probabilities.

Our class this morning provided a more in-depth look at probability, and today’s material focused most on a topic called Markov Chains. Essentially, a Markov Chain is a series of phases in which the outcome of each phase is one of a set number of possibilities, and the probability of a certain outcome occurring given the event of a prior state can be determined exclusively from the data from these two phases.  A Markov Chain qualifies as a stochastic process, meaning that at each phase of the chain there is more than one possible outcome, and the chosen outcome is variable.  


For example, this can be more easily visualized by studying a tree diagram; with this type of display, the tree begins with a single state (or event) and has a set number of offshoots of different possible states (see picture).  The probability that one event occurs given the previous state is calculated by multiplying the probability values along the two branches (this is known as conditional probability).  Additionally, we were able to utilize matrices in order to learn how to work with Markov Chains that have more states than can be easily displayed on a tree diagram, and we learned that matrix multiplication can be used to find the conditional probabilities of a Markov Chain.  Questions that Markov Chains may ask are ones like: “Given the fact that a person rides Rock ‘n' Roller Coaster once, what is the probability that that person gets back in line for the ride? What is the probability that they choose to ride another particular attraction in the park?”  These are the types of ideas Markov chains can describe and matrices can help us to solve!    

Genetic Algorithms

As the genetic project presentations are continuing to approach, each group spent several hours making progress on their respective projects this morning after class. To complete this project, we are using the same data that we were given for the traveling tourist problem and are asked to meet the same basic objective; however, this time we only need a path that completes 10 rides in the shortest time instead of 19 attractions. The time it takes to complete the path is called the "tour fitness", and the goal is to find a tour with as minimal a time as possible to complete all the attractions. The project requires that we first choose a set number of parent tours that represent different paths that could be potentially used to complete the 10 rides. We then use different operators to mutate a parent or cross over two parents to come up with a new child tour. Ideally, the child will have a better fitness than at least one of the parents in the population, and it will be able to replace the parent in the new population. After performing 30 mutations or crossovers (called iterations), we will have achieved a population of paths of the best fitness we have calculated, and the goal is to have a population of more fit tours than the ones with which we began.

Data Collection in the Parks   

There is always time for data collection in the parks, even on
Rock 'n' Roller Coaster!

As many of us began to express interest in using our own data for our future projects, the professors thought that today would be a good time to have us begin collecting data in the parks. For our final project, students will have the option to either work on a project involving queuing theory or a project of their own choosing. Based on the ideas that we brainstormed this morning, the professors decided that we would spend part of the day in Hollywood Studios instead of only in Epcot as we had previously planned.

Some of us have expressed interest in a project involving the fastpass line. How many people with fastpasses do Disney employees let onto the rides in comparison to the number of people in the standby line? Is it variable to the ride, the employee, or the time of day? To begin collecting data for this project, as we neared the front of a line, we attempted to count the number of people from each line that the employees let into the next segment of the line.  

Another set of students wants to look into how accurate Disney’s posted wait times are. Therefore, for each attraction we rode we recorded the ride, the time of day, the type of line, the posted wait time, the time we entered the line, and the time we got on the ride.

A third group of people is interested in the single rider line and measuring how beneficial the single rider line is for each of the rides that offer it. Test Track in Epcot, Rock ‘n’ Roller Coaster in Hollywood Studios, and Expedition Everest in Animal Kingdom all utilize a single rider line.

Epcot

The simulator at Sum of All Thrills that visitors ride in order
to experience the coasters they have designed.

After project time at the hotel, our group loaded the van and headed to Epcot for the afternoon. As stated, the time in the park provided a valuable opportunity to begin the data collection for our upcoming projects.  Because official groups haven’t yet been decided on for each project, we all worked collectively on gathering data throughout the day to help with all of the potential future projects.  For example, I (Maria) worked on gathering data about single rider lines; specifically, at Test Track (and later at Rock ‘n’ Roller Coaster), I observed the group size of people that were let onto the rides consecutively (which is usually called out by the worker heading the line) and how often single riders are asked to board the ride.  I also worked on collecting data for the project that seeks to compare posted wait times with actual wait times; with a group of several other people, I visited multiple attractions in order to time my wait in each queue and compare them to the posted times.


During our time at Epcot, we were also able to visit an attraction called Sum of All Thrills for the first time, which involves pairs of people designing their own rollercoasters on a computer and entering a simulator to ride the coaster they have designed.  This provided a perfect example of the prevalence of math and science in the park, and as the ride voiceover reminded us before we worked on our designs: “Control the math, control the ride!”  When my group (Courtney) was designing our rollercoaster, we were able to choose what types of thrills we wanted the track to do, the height of the track, as well as the speed. A small crash robot on our design screen would test the ride to make sure it was safe. After our first try, the robot reminded us that our speed had to be compatible with the height of the track. We manipulated our design for another minute and discovered that the higher we made the track, the faster the coaster needed to travel. It is exciting to see how Disney makes an effort to not only give children a fun experience on a ride, but also make it an educational experience.


Before leaving Epcot, we got to eat dinner in the World Showcase - the variety of cuisine among all the countries in the park always makes for a delicious event!

Hollywood Studios

After our Epcot dinner, we reconvened as a group and headed toward Hollywood Studios. This is the first occasion that we have visited two parks in the same day, and the change provided a good way to continue to diversify our data collection.  Our group spread throughout the park and were able to visit and collect information from a number of attractions.  I (Maria) visited Rock ‘n’ Roller Coaster twice over the course of the evening; the first time, I went with a large group and we spread throughout the line in order to broaden the amount of consecutive data we could collect.  As explained above, we focused on recording the group sizes called onto the ride; by spreading ourselves throughout the ride, we could collect information about group sizes for a long line of people.  Many people from the group were also able to visit Tower of Terror, Star Tours, a Frozen Sing-Along - and Dr. Hutson even reached a new high score at Toy Story Midway Mania!  Of course, no evening at Disney World is complete without a fireworks show, and our group enjoyed getting to see the impressive Star Wars-themed fireworks display before leaving the park.



Tomorrow, each group will present its genetic algorithm project results in the morning, and after class we’ll head back to Hollywood Studios to continue our data collection!

Written By:

Maria & Courtney