# Probability

List of probability topics:

- https://en.wikipedia.org/wiki/List_of_probability_topics
- https://en.wikipedia.org/wiki/Outline_of_probability

P(x,y) is the same as

`P(x \cap y)`

A very common notation that you need to understand is that $P(x,y)$ is the same thing as saying $P(x∩y)$.

Learned this to learn about how we can create AI that makes optimal decisions given limited information and uncertainty.

The classical definition of probability: $P(A)=# of elements in S# of elements in A $

However, there are 3 problems with this classical definition:

- Logically inconsistent
- Elements in $S$ may be difficult to count
- $S$ needs to be finite (in the case for example of infinite sample space)

There are two extended interpretations of probability:

4-Part Method

- Identify the sample space (i.e. outcomes) → helps to use a tree
- Define Events of Interest (ex: when you win)
- Determine Outcome Probabilities
- Assign Edge Probabilities
- compute Outcome Probabilities

- Compute Event Probabilities

Variables vs. Events?

**Unconditional Probability**
Unconditional probability is the degree of belief in a proposition in the absence of any other evidence. The result of rolling a die is not dependent on previous events.

### Concepts

- Axioms of Probability
- Conditional Probability
- Bayes Rule
- Joint Probability
- Probability Rules
- Law of Total Probability