From d69bb7c250558d0121d8eb549fe68594cff96ccd Mon Sep 17 00:00:00 2001
From: Holden Rohrer <hr@hrhr.dev>
Date: Tue, 18 Aug 2020 15:17:01 -0400
Subject: added notes (p much a transcription) for lecture one

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+Probability is the study of randomness.
+
+Random Experiment:
+    Ex: Flip a coin, roll a die, results of a Covid test, ...
+    - It is random because the outcome is uncertain
+    - Can list all possible outcomes
+
+Defun 1: The sample space associated with an experiment, Omega, is the
+set of possible outcomes of an experiment.
+
+Ex:
+    Flip a coin: Omega = {heads, tails} | {heads, tails, sides}
+    Roll a seven-sided die = {1..7}
+        The experiment, rigorously, is the # of dots on the top face.
+
+Cardinality of set is number of elements of set.
+Cardinality of empty set is 0. (why is this a convention and not a
+self-evident fact?)
+
+Card S = #S = cardinality of S
+    
+    Flip a coin until first tail, and stop. Count # of flips.
+    Omega = {1,2,3,...} = Natural numbers. (Sometimes also {0,1,2,...})
+Card N = +\infty
+
+Suppose you flip a coin and want to assess the odds of getting tails on
+a flip. For any given coin, we don't know. How can we figure out?
+One possibility: Keep flipping coin. # of tails / # of flips is
+intuitively close to the probability of getting tails.
+T(n) = # of tails / # of flips.
+Intuitively,
+0 <= Omega(n) = T(n)/n <= 1.
+For n large, Omega(n) ~ probability of tails on a single flip.
+
+The law of large numbers tells us that T(n)/n will converge as
+n -> +\infty, to p---the probabiity of getting tails on a single flip.
+
+"Wisdom of Crowds" somehow obeys the law of large numbers.
+
+This is the "frequentist" approach for probability theory
+
+"Subjective approach" to probability theory
+    - Better at answering question like "what is the odds that Dow Jones
+      grows by 113 points by EOD?"
+
+d) Pick a point at random in the unit square and observe its coordinates
+    What is the probability space?
+    Omega = [0,1] x [0,1] = [0,1]^2 = {(x,y) : 0<=x<=1 : 0 <=y<=1}
+        -  Could also have excluded the boundary, making it (0,1)^2
+    #Omega = +\infty
+
+Assuming a uniform distribution:
+The odds of choosing a point in the bottom left corner triangle is 1/8.
+The odds of choosing a circle with a radius of 1/4 is pi/16.
+Generally, area/total = odds.
+Point area = 0, so odds of choosing it is 0.
+
+How to formalize the uniform distribution?
-- 
cgit