Editing Sample space (section)

== Conditions of a sample space ==
A set <math>\Omega</math> with outcomes <math>s_1, s_2, \ldots, s_n</math> (i.e. <math>\Omega = \{s_1, s_2, \ldots, s_n\}</math>) must meet some conditions in order to be a sample space:<ref name="conditions">{{cite web |url = https://ocw.mit.edu/resources/res-6-012-introduction-to-probability-spring-2018/part-i-the-fundamentals |title = Sample Spaces |last=Tsitsiklis |author-link=John Tsitsiklis |first=John |date=Spring 2018 |publisher=Massachusetts Institute of Technology |access-date=July 9, 2018}}</ref>
* The outcomes must be '''mutually exclusive''', i.e. if <math>s_j</math> occurs, then no other <math>s_i</math> will take place, <math>\forall i,j=1,2,\ldots,n \quad i\neq j</math>.<ref name=":0" />
* The outcomes must be '''collectively exhaustive''', i.e. on every experiment (or random trial) there will always take place some outcome <math>s_i \in \Omega</math> for <math>i \in \{1, 2, \ldots, n\}</math>.<ref name=":0" />
* The sample space (<math>\Omega</math>) must have the '''right granularity''' depending on what the experimenter is interested in. Irrelevant information must be removed from the sample space and the right [[abstraction]] must be chosen.

For instance, in the trial of tossing a coin, one possible sample space is <math>\Omega_1 = \{H,T\}</math>, where <math>H</math> is the outcome where the coin lands heads and <math>T</math> is for tails. Another possible sample space could be <math>\Omega_2 = \{(H,R), (H,NR), (T,R), (T,NR)\}</math>. Here, <math>R</math> denotes a rainy day and <math>NR</math> is a day where it is not raining. For most experiments, <math>\Omega_1</math> would be a better choice than <math>\Omega_2</math>, as an experimenter likely does not care about how the weather affects the coin toss.