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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?
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