# Lec 1

Experiment with uncertain outcomes.

We would like to measure the likelihood of an outcome or a group of outcomes.

* How to make this rigorous.
* Random numbers.

Repeated experiments (patterns)

Modeling real life examples

Flip a fair coin: probability of head? $$\frac{1}{2}$$

Roll a fair die:  {3,6} = p = $$\frac{2}{6}$$

1. flip a coin 10 times, probability of getting 10 tails.
2. Flip a  coin until get a tail. What is the probability we need to flip coin more than 5 times.
3. Suppose you have a dart board. Radius of 9 inches. What is the probability of landing less than 1 inch

Answer

1: $$\frac{1}{2}^{10}$$

$$(a\_1, a\_2, \dots, a\_{10})$$, $$a\_i \in {H, T}$$. There are 2^10 outcomes. One would be all tails. So $$\frac{1}{2^{10}}$$.

2: first 5 flips are all heads. $$\frac{1}{2^5}$$. 

3: $$\frac{\pi}{81\pi}$$= $$\frac{1}{81}$$. 

Experiment:

Probability of something. 

Kolmogorov's axioms of probability.

$$\Omega$$ sample space. Set of all possible outcomes. (Non-empty, at least 1 possible outcomes)

$$F$$: set of event. Event is a subset of $$\Omega$$.  $$\emptyset \in F, \Omega \in F$$

$$P$$: function $$f \rightarrow \[0,1]$$ . (Probability of the event)

$$(\Omega, F, P)$$ - probability space.

$$F \subset {\text{all subset of} \Omega} = 2^{\Omega} = P(H)$$

E.g

Flip a coin.  $$\Omega = { H, T}$$ $$F = { {H}, {T}, \emptyset, {H, T}}$$

Roll a die $$\Omega = {1,2,3,4,5,6}$$ 

Flip a coin 10 times.$$\Omega = {(a\_1,\dots,a\_{10}); a\_i \in {H,T}}$$$$= {H,T}^{10}$$

$$A, B$$ sets. $$A \times B = {(a,b), a\in A, b\in B}$$

$$A \in F$$ $$0\leq P(A) \leq 1$$. 

$$P(\emptyset) = 0$$

$$P(\Omega) = 1$$

$$A\_1, A\_2, \dots,$$ disjoint event. $$P(\bigcup\_i Ai) = \sum\_i P(A\_i)$$. (Additivity)


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://tianyi0216.gitbook.io/blog/course_notes/431-notes/lec-1.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.