Overview

Probability is cool! Cause humans are bad at it without math

For this class, we have to do the reading ahead of time, since Keith does not lecture. There are lots of cool lectures online if you need help on the more basic stuff, but Keith likes the cool stuff.

Learning goals,

1. Learn the axioms
2. Learn to compute simple probabilities/distributions
3. Random variables
4. Bayes’ Theorem
5. Conditional probabilities and expectations
6. Significance of independent random variables
7. Weak Law of Large Numbers, Strong Law of Large Numbers, the Central Limit Theorem

What the f--- is probability?!
Ideas,

1. Multiverse
2. Limit as trials goes to infinity
3. Weather reports????

Philosophically, we call #2 frequentism, since it refers to the relative frequency of an event

(FYI, Keith recommends MATH 36B, which requires more thought towards this question.)

When someone says, “the chance I get a raise is 60%,” how do we interpret this through the definition of frequency? An alternative philosophy is that the statement reflects the speaker’s beliefs, their level of confidence.

In statistics, confidence is a key concept. In this class, the difference is not as important, since the pure math does not care much.

A probability space assigns probabilities to outcomes of an experiment. It is defined with the items $(\Omega ,P)$. For any $A\subseteq \Omega$, our probability space defines $P(A)$, where $0\le P(A)\le 1$.

So say we roll a d6, $\Omega =\left\{1,2,3,4,5,6\right\}$
Let’s define two events, $A=\left\{3\right\}$ and $B=\left\{1,3,5\right\}$
We know very little about this situation (if we don’t assume the dice is fair), however we can say with certainty that $P(A)\le P(B)$.

This is an example of monotinicity; as outcomes are added to an event, the probability increases or stays the same. When $A\subseteq B$, $P(A)\le P(B)$.

Probability spaces also showcase disjoint additivity, where the probability of a sum of a collection of disjoint sets is equal to the sum of their individual probabilities. FYI, this is only valid on countable sets.

Also, $P(\varnothing )=0$ and $P(\Omega )=1$

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