# 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. $$\frac{1}{2^5}$$.&#x20;

3: $$\frac{\pi}{81\pi}$$= $$\frac{1}{81}$$.&#x20;

Experiment:

Probability of something.&#x20;

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 $$\Omega = {1,2,3,4,5,6}$$&#x20;

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

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

$$A \in F$$ $$0\leq P(A) \leq 1$$.&#x20;

$$P(\emptyset) = 0$$

$$P(\Omega) = 1$$

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