SP18:Lecture 31 Probability intro
We'll start discussing probability, giving motivation and basic definitions.
Probability is a mathematical tool for estimating the "likelyhood" of various events. It can be used to reason about uncertainty or lack of knowlege, giving a framework for integrating different assumptions and observations.
Sample space, event, outcome
Probability measure, Kolmogorov's axioms
Facts about probability
Everything else about probability follows from the Kolmogorov axioms. For example:
Rearranging this equation gives the result.
In particular, we have:
Example: six-sided die
The proof is left as an exercise.
Note that by choosing this sample space, we are already ruling out the possibility that the die could land on a corner or roll off the table; it is important to be aware that the choice of model can affect the conclusions drawn using it.
Example: sum of two dice
Either sample space would work, since both contain enough information to describe the outcomes of the experiment. However, it is difficult to describe the probability measure with the first model, while it is easy with the second (for the second, the fact that the dice are fair and independent means that the equiprobable measure is a good probability measure to describe the experiment).
There is not a "correct" sample space for a given problem, but there are some that are easier to work with than others. It is also possible to create a sample space that doesn't have enough resolution to interpret the events of interest: for example the sample space wouldn't work for this experiment, since there is no way to interpret the sum of the dice.
Example: height of random person
Suppose we wished to model the following experiment: select a person from the room (uniformly), and measure their height.
The former choice is more difficult to set up, because figuring out the probability of selecting a given height requires knowlege of the heights of the people in the room; choosing the latter is easier because one can use an equiprobable measure.