Probability defined (and figuring out how to host IPython notebooks)
Hosting notebooks
Got side tracked for a bit figuring out how to render notebooks for viewing on this site but I'm pleased with the result. It's nice to not have to point to nbviewer.jupyter.org, which seems to have quota issues from time to time. Aside from lektor specific issues, the best way to host a rendered notebook within an existing layout on your site is:
- Render the notebook using the jupyter nbconvert command, e.g
jupyter nbconvert --to html --template basic mynotebook.ipynb - Include a stylesheet to make the notebook look nice. I cribbed the one from the nbviewer website.
In any case, I think notebooks will be a nice way build up some longer lasting resources that are more centered around concepts than day to day progress. On a related note, I just came across Peter Norvig's Introduction to Probability iPython Notebook and look forward to working through it and to perhaps produce similar notebooks as I learn different concepts.
Probability defined
Continuing here in stanford course, the section, "Finding probability of events".
Note: this also overlaps some of the material from chapter 1 of "All of Statistics", I'm enumerating all terminology below.
- Ω: Sample space
- ω: outcome or point in Ω
- A: an event, which is a subset of Ω, or a set of ωis
Example: tossing a coin three times.
Ω = {HHH, THH, HTH, HHT, HTT, THT, TTH, TTT}
Some example events within this sample space:
- There are no heads: {TTT}
- Getting exactly one head: {HTT, THT, TTH}
- Getting at least one head: {HHH, THH, HTH, HHT, HTT, THT, TTH}
Probability of outcomes and events
Assuming equally likely outcomes, the probability of an event is |A| / |Ω| or the number of outcomes in A divided by the total number of outcomes.
So the probability of getting exactly one head is 3/8.
It's not always easy (or can be tedious) to enumerate every outcome in a sample space; this is where counting methods can come in handy. For example, the number of outcomes in the sample space, "chose two cards from a deck" is equal to 52 * 51, since you have 52 choices the first time and 51 the second.
Also, It's not always the case that outcomes are equally likely but there are some rules that are always true concerning events and probabilities.
- the probability of any given outcome is between 0 and 1
- the sum of the probability of the outcomes must add up to 1
- the probability that an event will not occur is 1 - P(A)
Independent (disjoint) events
Two events are disjoint if they do not share any outcomes.
If two events are disjoint:
- P(A∪B) = P(A) + P(B)
- P(A∩B) = P(A) * P(B)