202405240847
Status: #idea
Tags: Probability Theory
State: #nascient

Event

Informally, an event is any subset of of the sample space $\Omega$ which is of interest to the experimenter. If we take a coin toss, technically it is possible for the coin to fall on its side and that neither tail nor head show up. This would be a possible outcome (that we might, or not account for) but not an event since we don't care about it. Rigorously tho an event is any element of the $\sigma -$algebra that is equipped by our $\Omega$. Simply because any measure we could muster will only work on that $\Omega$ and anything else cannot be measured.

An event will often be worded similarly to a filter in programming languages. Like we are trying to capture a subset of the data and think about its specific properties. ie: We are looking for the outcomes that are even.

event_even = df.loc[df['count_of_heads'] % 2 == 0] # checking for even

Consequences of this definition

• If something occurring is of interest, then it not occurring is also of interest (it's the dual of it in a sense)

# if we try to create a event_even, then
even_even_prime = df.drop(event_even.index) # the odd subset is created at the same time

• If two events are occurring? Then their interaction with each other will be of interest as well. What is their union? their intersection?
• The sample space is always an event by definition, it is literally the thing we are working on. Our workbench so to speak.

These altogether give us what's referred to as an Algebra.

Independence of Events

We say two events $A$, $B$ are independent when:
$P(A\cap B)=P(A)P(B)$

If $A_1,A_2,A_3,...$ are independent then any combination of elementary set operations on those sets will be independent (assuming obviously you do not reuse sets.)

A set $A$ is independent of itself in only two cases by the definition, either $A$ is $\Omega$ or $A$ is $\varnothing$.

By that same definition, except in the aforementioned cases, $A$ and $A^c$ are never independent. And after all this makes perfect sense, informally independence tells us that knowledge about one event occurring gives us no information about the other occurring. But here $A$ and $A^c$ are quite the opposite, in fact they are perfectly dependent. $A$ occurs iff $A^c$ does not!

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