Week 3: Research on Mathematics / by Valzorra

Chance and Probability

The simplest possible mathematical definition of Probability would be the chance of a certain event taking place. As we have covered the basic definitions and methodologies behind Chance and Probability in James’s Tech Workshop and Building the World Sessions, I will proceed on straight to more advanced concepts we have not gone over. If you would like a more basic overview of Probability, please refer to Week 1: A Welcome Return and Week 2: Calculus and Cancer.

Die are a fantastic way for obtaining a random number. This connects to my research in Geometry as die are only produced in the shapes of the Platonic Solids and another set of solids known as Fair Die.

Die are a fantastic way for obtaining a random number. This connects to my research in Geometry as die are only produced in the shapes of the Platonic Solids and another set of solids known as Fair Die.

Fair Dice

The concept of Fair Dice is quite exciting to me as that’s a meeting point between the study of Chance and Probability and Geometry. In mathematics, Fair Dice are solids which offer an equally likely chance to land on any one surface of the object. Following that definition, all of the Platonic Solids can be considered Fair Dice, including the most popular dice, the Cube (for more information, please refer to Week 3: Research on Geometry). However, there are a variety of shapes that are not necessarily Regular Polyhedrons, but can be used as fair dice. The main requirement for dice to be considered fair is that the overall geometric shape of the dice must remain unchanged and symmetrical, regardless of which face it is landed on. If we were to explore the concept in even more detail, we would also have to consider how the real world affects the way die land, and whether or not it is possible to influence its tumble, however, that’s a related to physics rather than pure mathematics. What’s exciting about Fair Dice is the variety of probabilities they offer, more so than a traditional six-sided one. If I were to design a Board Game, I would look into a variety of Fair Dice, and I would use the Binomial Distribution Formula from my prior research to estimate what odds would be most appropriate for my design. For more information on Fair Dice, I highly recommend the video below by Professor Persi Diaconis from Stanford University, who is an expert in the field.

The Monty Hall Problem

A fascinating problem surrounding Chance and Probability is the so-called Monty Hall problem. Monty Hall was a TV Presenter of the popular Quiz Show Let’s Make a Deal. The simple premise of the show is that contestants are presented with three doors, one of which conceals an impressive prize, while the other two do not contain anything of value. For ease of reference, let’s note the three doors as A, B, and C. If a contestant has selected door B, for example, they have a 1/3 chance of hitting the jackpot at that point. After the contestant’s first selection, Monty Hall does not give them the results straight away. Instead, the TV presenter reveals another door, Door A for example, which contains nothing valuable. What’s crucial here is that Monty knows where the jackpot is, and he purposefully shows the contestant a door with nothing of value. After this has been revealed, the player is then asked whether they would like to change their mind about Door B and opt for Door C, instead.

Monty Hall during a section of the game show.

Monty Hall during a section of the game show.

At that point, one would naturally assume that now the odds of Door B containing the prize have increased to 1/2, as we have been left with only two options to choose from rather than the initial three. However, the counter-intuitive answer is that actually, there is a 2/3 chance of Door C containing the jackpot and only a 1/3 chance of Door B being the correct one. Therefore, when presented with this situation, the player should always choose to change their mind from their initial selection. The reason for this answer lays in the knowledge Monty Hall possesses of where the prize actually is. Since he knows where the prize is in advance, he knows which door to reveal to the player, so that he does not unveil the jackpot. One way to look at it involves assuming that we will always switch as a player, which leaves us with the following three options:

  1. Select Door A - Door A is the correct answer - One of the two remaining empty doors has been revealed - We switch - We loose as Door A was correct all along.

  2. Select Door A - Door B is the correct answer - The only other empty door, Door C, has been revealed - We switch - We Win.

  3. Select Door A - Door C is the correct answer - The only other empty door, Door B, has been revealed - We switch - We win.

If we were to always remain with our original choice, there is only a 1/3 chance of us being correct, as we were originally presented with the three options. The change in circumstances and statistical probability occurs once one of the empty doors is revealed. These results are purely the cause of Monty Hall’s knowledge of the contents of each door. If he was unveiling them at random, and happened to unveil an empty door, then the chance of either of the remaining options containing the prize would in fact be 50/50, which is what intuition suggests.

Thoughts and Reflection

The Monty Hall problem illustrates a key aspect of Chance and Probability that greatly interests me, which is the manipulation of chance, and how we as designers can have absolute power and control over these circumstances. Mathematical formulas such as the Percentile Mechanic and the Binomial Distribution Formula allow us to calculate chance and then manipulate it to fit our needs or make educated decisions about how to handle a situation. Knowing these principles gives us incredible control as creators and allows us to grab a hold of our universe and make it fit our design as we please. Controlling chance and probability has immense power, and at that point it can even connect to the philosophical debate of fate against free will. An interesting question to consider would be if fate and destiny can possibly exist in a universe where one has control over coincidences and chance events. An interesting follow-up question would be whether fate can be defined as pure chance or if there is more to the concept than that.