Table of contents

1. Concepts
   1. 1. Random experiment
   2. 2. Sample space, Event
   3. 3. Probability
2. References

Concepts

1. Random experiment

Studies and experiments are conducted to deal with the uncertainty by estimate the likelihood of a specified event (e.g. two tossings produces two heads).

Random experiment. A process (or mechanism) produce definitive but uncertain output (or outcome) (e.g. tossing a coin).

Random experiments are often conducted repeatedly, so that the collective results may be subjected to statistical analysis. A fixed number of repetitions of the same experiment can be thought of as a composed experiment, in which case the individual repetitions are called trials. (Experiment - Probability Theory, 2021)

Some characteristics of a random experiment includes:

• consists of multiple trials (tossing the coin multiple times). Each trial produces one and only one outcome (e.g. head or tail)
• a random experiment has a set of all possible outcomes, called Sample space, denoted as $\Omega$ or $S$ (e.g. $\Omega = \{Head, Tail\}$)

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2. Sample space, Event

The finest-grained list of outcomes for an experiment is the sample space of the experiment (Larson & Odoni, 1981)

Sample space $(\Omega _{[omega]}$ or $S )$ . A set of all possible outcomes associated with a Random experiment. Each Outcome is an element of the Sample space.

Sample space’s cardinality (i.e. size of a set)—represented as $n(A)$—could be ‘finite, countably infinite, or noncountably infinite’.

To summary, there are 3 different types of Sample spaces:

Events. A group of zero or more outcomes. All events is a member of $\mathcal{F}$ (\mathcal{F}) which is a set of all events.

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Editing

• Key operations details
• Algebraic details

Algebra of Events:

3 key operations

Union

Intersection

Complement

7 algebraic axioms

Commutative law

Associative law

Distributive law

Complement of the complement

Complement of the intersection

Intersection of self-complement

Intersection with the universe of events

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3. Probability

Back to aforementioned statement in the previous section:

Studies and experiments are conducted to deal with the uncertainty by estimate the likelihood of a specified event.

So where does the uncertainty in the event occurence come from?

According to (Shapiro, 1999), there are 3 kinds of source leading to the random behavior in a [classification] system:

1. True random component, e.g., random noise in measurement
2. Random component due to modeling, e.g., systems learn from sampled data
3. Incomplete information, e.g., unknown domain knowledge or knowledge on the phenomena

To address these uncertain elements, we assign a Probability—a value between 0 and 1—to an event.

Probability. Likelihood of each of the possible events (on a single trial), denoted as a continuous number in range $[0, 1]$. $\mathbf{P}$ is a function mapping from events to probabilities.

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Axioms of Probability:

1. Axiom 1: For any event $A, \mathbf{P}(A)\geq 0$.
2. Axiom 2: $\mathbf{P}(S) = 1$, $S$ is the Sample space
3. Axiom 3: Suppose $A_{1..n}$ are disjoint events: $\mathbf{P}(A_1 \cup A_2 \cup ...) = \mathbf{P}(A_1) + \mathbf{P}(A_2) + ...$

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References

1. Experiment - Probability theory. (2021). https://en.wikipedia.org/wiki/Experiment_(probability_theory)

   [Online; accessed 2021-07-14]
   LINK
2. Larson, R. C., & Odoni, A. R. (1981). Brief Review of Probabilistic Modeling. In Urban Operations Research. Prentice-Hall. https://web.mit.edu/urban_or_book/www/book/chapter2/contents2.html

   [Online; accessed 2021-07-14]
   LINK
3. Shapiro, J. (1999). A Primer on Probability. https://apt.cs.manchester.ac.uk/ftp/pub/ai/jls/CS2411/prob97/

   [Online; accessed 2021-07-15]
   LINK