Explain what is meant by the term sample space
Explain why we introduce a set of all possible subsets of the sample space.
Give the axioms that define the probability measure
What interpretations of probability are possible?
What is the classical interpretation of probability and what are its limitations?
Imagine an experiment performed in the future. The outcome from this experiment, A, is something that we cannot possibly know in advance of doing the experiment. We might know at the very least however we know that A will be a one from amongst a set of mutually exclusive outcomes - the sample space $\Omega$.
We now define a second set, $\mathcal{F}$, that contains all possible subsets of the sample space $\Omega$. We can then define a probability measure (or probability distribution) $P$ that maps each element of $\mathcal{F}$ to the set of real numbers in the closed interval $[0,1]$ as follows:
$$ P : \mathcal{F} \rightarrow [0,1] $$
The axioms that define $P$ are:
$$ \begin{aligned} & P(\emptyset) = 0 \\ & P(\Omega) = 1 \\ & 0 \le P(A) \le 1 \quad \forall \quad A \in \mathcal{F} \\ & \textrm{if} \quad A\cap B = \emptyset \quad\textrm{then} \quad P(A\cup B) = P(A) + P(B) \quad \end{aligned} $$
The theory of random variables and the cumulative probability distribution function is constructed using the formal definition of a probability measure given above. We can only use a single random variable, however, if we know that all the possible outcomes from our experiment are mutually exclusive.
It is important to note that the interpretation of this quantity called probability is contentious. There is much dialogue between the so-called frequentist and bayesian interpretations of this quantity.
The simplicity of the classical interpretation that is explained in the video below is, however, is useful when solving some problems. This explanation is incomplete, however, as it cannot be used if the sample space contains an infinite number of outcomes.
https://www.youtube.com/watch?v=dEzLR-tReEY
<aside> 📌 SUMMARY: The set of all possible outcomes of an experiment is called the sample space. We define the probability measure as a function that maps each subset of the elements in the sample space to a number between 0 and 1.
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