This is going to be harder than I thought...
Probability is the logic of uncertainty.
The gist of it is that probability is used in a wide variety of fields: statistics, science, computer science, political science, etc. He makes the exact same joke in his lecture as he does in the book about mentioning finance and gambling being repeating himself.
It mentions that they will be using simulations as a way to reinforce ideas from the book. I’m looking forward to that. I’ve always been fasninated with randomization and simulation. I’ve got websites from 15+ years ago when I was teaching myself javascript where I wrote lottery simulations, poker simulations, coin-flip simulations, etc. Actually, the very first full-fledged program that I wrote myself, as a final project for Super Saturday at Purdue when I was in 6th grade (25 years ago) was a coin-flipping simulation written in BASIC. The simulations will be focused on R, a programming language that I’m already pretty good at so it should be fun.
This section introduces sets and defines a few new terms related to sets. It then uses several examples to reinforce these concepts.
The pebble world is a visualization technique that encourages you to look at the sample space as a world populated by a bunch of identical pebbles. In this world, each pebble is equally likely to be selected and represents an potential outcome. That makes an event be a group of pebbles. The ratio of pebbles in an event to the total number of pebbles becomes the probability of that event occuring. This visualization requires that two assumptions be true:
The section goes on the introduce some additional nomenclature:
~(AuuB)^c=A^cnnB^c~; ~(AnnB)^c=A^cuuB^c~
This law says that the compliment of the union of two events is the same as the intersection of the two events’ compliments. Similarly, the compliment of the intersection of two events is the same as the union of the two events’ compliments. Now to try to put this is terms that actually make sense to me.
We will stick with ~A~ being drawing a Club from a deck of cards and ~B~ being drawing an Ace. ~AuuB~ would be drawing any Club or Ace. The compliment of that would be drawing any non-Club that also isn’t an Ace. ~A^c~ would be drawing any non-Club. ~B^c~ would be drawing any non-Ace. The intersection of those would also be any non-Club that is also a non-Ace.
For the second one: ~AnnB~ would be drawing the Ace of Clubs. The compliment of that would be drawing anything but the Ace of Clubs. Again, the compliment of ~A~ would be drawing a non-Club and the compliment of ~B~ is drawing any non-Ace. The union of those would cover drawing anything but the Ace of Clubs.
New symbols are introduced here. For example,
~B=uuu_(j=1)^10 A_j~
Given that ~A~ is a coin flip resulting in Heads, The above indicates that ~B~ is the event that any of 10 coin flips is heads. It works just like summation ~sum~, its the union (~uu~) of the ten events. Similarly,
~C=nnn_(j=1)^10 A_j~
says that ~C~ is the event that all ten coin flips result in heads. The last example is a bit more confusing:
~D=uuu_(j=1)^9 (A_jnnA_(j+1))~
says that ~D~ is the event that at least one set of consecutive coin flips will be two Heads. There are only 9 iterations this time because the 10th is taken care of by the 9ths ~A_(j+1)~. It is the union of 9 intersections with each intersection’s first event being the second event of the previous intersection (except, of course for the first intersection). I think I understand this, though I don’t think I’ve explained it very well.
He is really big on it being important to be able to use English to explain what is going on (Story Answers) and to that end he provides a table at the end of Section 1.2 that is meant to aid in translation between English and Set Notation. I considered repoducing the table here but I’m not sure what the copyright implecations of that would be.
The book defines the naive definition of probability as counting the number of ways that an event could occur and dividing that by the number of possible outcomes. This is a very restrictive definition because it assumes that all outcomes are equally likely and that there are a finite number of outcomes. This naive definition seems to be the basis of the Pebble World that is mentioned in the previous section. It is defined more formally as:
~P_(naive)(A) = (|A|)/(|S|)~
where ~A~ is an event in the finite samples space ~S~ and ~|A|~ is the total number of outcomes that would result in ~A~ and ~|S|~ is all possible outcomes in the sample space.
He once again recycles a “joke” (switching Mars for Neptune in the lecture) to try to illustrate the problems with the naive definition. The book goes into much more detail about when the naive definition is acceptable and when its not. I think this is pretty straight forward.