Lecture 1

Random experiment a process leading to a number of possible outcomes with actual outcome depending on influences which cannot be predicted

Sample space \(\Omega\) contains all possible outcomes

Outcome a sample point \(\omega\) that belongs to the sample space \(\omega\in\Omega\)

Event a set of possible outcomes, which is a subset of the sample space \(A\subseteq\Omega\)

Probability a function \(\mathbb P\) which is \(\mathbb P\in[0,1]\) and assign the value to every event on the sample space

thus we have: Let \(N(A)\) be the number of outcomes belong to event \(A\)m thus the probability of \(A\) would be \(\mathbb P(A)=\frac{N(A)}{N(\Omega)}\) which strictly requires \(N(\Omega)<\infty\) and all possible outcomes have the exact same likelihood of occurring

Certain event, \(\Omega\) itself, which have a probability of 1

Impossible event, \(\emptyset\) that have a probability of 0

Complement \(A^c=\{\omega\not\in A|\forall \omega\in\Omega\}\)

Set difference \(A\backslash B=B^c\cap A\)

Subset (equal) which is denoted as \(A\subseteq B\)

Disjoint \(A\cap B=\emptyset\)

Exhaustive events if the union of them is \(\Omega\)

Distributive Laws: \(A\cap(B\cup C)=(A\cap B)\cup (A\cap C)\) and \(A\cup(B\cap C)=(A\cup B)\cap (A\cup C)\) De Morgan's Laws

De Morgan's Laws: \((A\cup B)^c=A^c\cap B^c\) and \((A\cap B)^c=A^c\cup B^c\)

Dice Roll Problem

Random experiment: Roll 2 distinct dice and record the sides they land on

Sample space: \(\Omega = \{(1,1),(1,2),(1,3),....,(1,6),(2,1),...,(6,6)\}\)

Outcome: any pair \((i,j)\) such that \(i,j\in\{1,2,3,4,5,6\}\)

Event \(A\) be the event such that the sum of the sides equal to 3

Probability of \(A\) \(\mathbb P(A)=\frac2{36}=\frac1{18}\)