#SUGT – Stanford University’s Game Theory course

Several weeks ago I signed up to receive updates on an innovative new set of online study courses run by Stanford University. I’d all but forgotten about the course until today, when I received an email informing me that the online resources for the Game Theory course were now open for registration, and the first video available to watch.

I’ll be spending the weekend watching that video and playing the example games in the Game Theory Lab; for now, I’m boning up on mostly forgotten probability theory mathematics with the help of Wikipedia. I’m taking notes in my trusty moleskine, but I’ll also use this blog to record what I’m learning. Here is Wiki’s plainspeak rundown of the relevant terminology:

Consider an experiment that can produce a number of outcomes. The collection of all results is called the sample space of the experiment. The power set of the sample space is formed by considering all different collections of possible results. For example, rolling a die produces one of six possible results. One collection of possible results corresponds to getting an odd number. Thus, the subset {1,3,5} is an element of the power set of the sample space of die rolls. These collections are called events. In this case, {1,3,5} is the event that the die falls on some odd number. If the results that actually occur fall in a given event, that event is said to have occurred.

Probability is a way of assigning every “event” a value between zero and one, with the requirement that the event made up of all possible results (in our example, the event {1,2,3,4,5,6}) be assigned a value of one. To qualify as a probability distribution, the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events (events that contain no common results, e.g., the events {1,6}, {3}, and {2,4} are all mutually exclusive), the probability that at least one of the events will occur is given by the sum of the probabilities of all the individual events.

The probability that any one of the events {1,6}, {3}, or {2,4} will occur is 5/6. This is the same as saying that the probability of event {1,2,3,4,6} is 5/6. This event encompasses the possibility of any number except five being rolled. The mutually exclusive event {5} has a probability of 1/6, and the event {1,2,3,4,5,6} has a probability of 1 – absolute certainty.

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