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)