Lec 2

Math 431 - Introduction to Probability

September 8, 2023

Last time.

Probability Space. A probability space is a triple $(\Omega, \mathcal{F}, P)$.

• $\Omega$ is the sample space.(set of all possible outcomes)

• $\mathcal{F}$ set of events. (Subset of $\Omega$). ($\emptyset, \Omega \in \mathcal{F}$)

• $P$ is a probability measure. $P: \mathcal{F} \rightarrow [0,1]$

$A_1, \dots, A_n$ disjoint -> $P(A_1 \cup \dots \cup A_n) = P(A_1) + \dots + P(A_n)$

$P(\bigcup_iA_i) = \sum_iP(A_i)$

Uniformly chose element froma finite set.

$\Omega$ is a finite set.

Each outcome is equally likely.

$\Omega = {\omega_1, \dots, \omega_n}$

$P({\omega_i}) = P({\omega_2}) = \dots = P({\omega_n}) = \frac{1}{n}$

$\sum_iP({\omega_i}) = P(\bigcup_i^n{\omega_i} =1$

$A\subset \Omega$. $P(A) = P({a_1, \dots, a_n}) = \sum_i=1^nP({\omega_i}) = \frac{|A|}{|\Omega|}$.

Worksheet.

1- Possible outcomes: ${(a,b,c), 1 \leq a \leq 6, b , c \in {H,T}}$

2- b. ${(a,b), 1\leq a, b \leq 6}$

d. $\Omega = {(a,b), 1\leq a \leq b \leq 6}$ - different. (1,1) is 1/36 and (1,2) is 1/18 (because (2,1) (1,2) is same.)

Sampling

We have a finite set $X$. Non empty set.

We take a sample of size $k$ from the set unfiromly at random.

Can take without replacement or with replacement.

Can also consider with order or without order.

With order: $(a_1, a_2, \dots, a_k)$.- ordered sequence. without order: pick 3 dice and get a set with 3 dice (without order information). - ${{a_1, a_2, \dots, a_k}}$

Sample from X with replacement k times.

$\Omega = {(a_1, a_2, \dots, a_k), a_i \in X} = X^k$

$|\Omega| = n^k$ , $n = |X|$

Sample from X without replacement k times with order

$1 \leq k \leq n$

$\Omega = {(a_1, a_2, \dots, a_k), a_i \in X, a_i \neq a_j \text{ if } i \neq j}$

$|\Omega| = \frac{n!}{(n-k)!}$

without replacement and without order

size = $k$

$\Omega = {{a_1, a_2, \dots, a_k}, a_i \in X, a_i \neq a_j \text{ if } i \neq j}$

$|\Omega| = \binom{n}{k} = \frac{n!}{k!(n-k)!}$