Problem of Points

Problem of Points is a classical problem in Probability.

The setting of the problem is a game played between two. Each round of the game has an equal likelihood of any player winning. It is agreed upon before the fame that the first player to win a certain number of rounds wins the game. The proceeds, equally contributed by both players, is given in its entireity to the winner.

Now, the question is, if the game were to be interrupted for some reason, how can the pot be divided fairly.

Pascal and Fermat worked on this problem to come up with a solution. The tenets of which is what forms the basis of expected value in probability.

To keep it simple, the solution involved calculating the odd of each player winning for the subsequent rounds that were to be played. And splitting the pot based on that. While Fermat provided a logical solution to compute the final split of the pot, Pascal came up with a solution that could easily compute the pot if they were to play a certain number of rounds.

This activity involved drawing up a decision tree to map out all possibilties and their odds in an efficient way. This gave birth to whats now known as the Pascals Triangle.

Egrodicity

Egrodicity is an idea where a point of a moving system will eventually have travelled through all parts of the space that the system moves in. Like smoke in room and it eventually filling up the room. The egrodic theorem has been used to describe common-sense phenomenon like mixing of liquids from a mathematical point of view. An egrodic process is a process that has the same time average and ensemble average. A time average of a process is the value of the process over an amount of time. While an ensemble average is the average of N number of identical process at a specific point of time. An egrodic process can be understood by looking at a snapshot of the process rather than the complete picture.

Monte Carlo Simulations

Monte Carlo simulations are used to determine the probability of an outcome from a model by using random variables. When the model contains a variable that is uncertain, Monte Carlo simulation takes that variable and assigns it a random value. Based on repeated runs of the simulation the end result is than averaged to provide an estimate. In many cases, this has proven to be more accurate than “gut feeling” and other soft methods. Since this model can output different outputs for the same input due to the random variable interference, it is a Stochastic Process. This method works when the model contains many coupled variables. The repeated simulation of the model can uncover patterns with varying inputs for the random variable.

Of course, a model can only predict and account for whatever is built into it. If there are inefficiencies and non-linearities that were approximated to simplify the model, that will be reflected in the outcome as well. Monte Carlo simulations have applications in a wide range of fields including finance, statistical physics, oil exploration etc.