A2 - probability defined on events

in a sample space $Ω$ , the probability of event $E$ is given by $P (E)$ , which is the probability that the outcome of the experiment lies in the set $E$

the three probability axions

P (E) \geq 0 \forall E

additivity: if $E_{i}$ is a countable sequence of events with $E_{i} \cap E_{j} = Φ \forall i \neq j$ , then

P (\cup_{i} E_{i}) = \sum_{i} E_{i}

P (Ω) = 1

P (E \cup F) = P (E) + P (F) - P (E \cap F)

P (E \cup F) \leq P (E) + P (F)

the above inequality only becomes an equality if $E$ and $F$ are mutually exclusive
consider throwing a dart equally likely to land anywhere in the interval $[0, 1]$
let $P (0.3) = ϵ > 0$ be the probability that it lands exactly at 0.3
then the probability of landing at any rational point would also be $ϵ$
since they are mutually exclusive events, their probabilities should add, which ends up being greater than one, contradicting $P (Ω) = 1$
thus, $P (0.3) = 0$
note that $P ([0, 0.3]) = 0.3$