Define set functions, sample space, and events

In order to understand the concept of probability, it is useful to think about an experiment with a known set of possible outcomes. This set of all possible outcomes is called the sample space \((S)\).

Sample space \((S)\) –set of all possible outcomes of an experiment.  

Example:If the experiment is “rolling a die”, then the sample space, \(S\), can be shown as

$$ S={ \left\{ 1,2,3,4,5,6 \right\} } $$

We can then define an event \((E)\) as any subset of a sample space or, in other words, any number of outcomes of an experiment.

Event \((E)\) – any subset of a sample space

Example: In the experiment “rolling a die”, we can define event \(E\) as “rolling an even number” and show event \(E\) as:

$$ E=\left\{ 2,4,6 \right\} $$

We may also be interested in looking at the outcomes of multiple experiments. Then, for any two events \(A\) and \(B\),

\(A\cup B\) is defined as the union of events \(A\) and \(B\), and includes all outcomes that are either in \(A\) or \(B\) or both \(A\) and \(B\). The union of \(A\) and \(B\) in the diagram below includes the entire shaded region.

\(A\cap B\) is defined as the intersection of events \(A\) and \(B\) and includes all outcomes that are in both \(A\) and \(B\). The intersection of \(A\) and \(B\) in the diagram below is shown as the darker shaded region and labeled as \(A\cap B\).

\(A’\) is defined as the complement of event \(A\) and includes all outcomes that are not in \(A\). \(A’\) is the darker region in the diagram shown below.

\(\emptyset \) is defined as the null event and does not contain any values. If \(A\cap B=\emptyset \) then \(A\) and \(B\) are said to be mutually exclusive. Events \(A\) and \(B\) in the diagram below are considered mutually exclusive.

If all outcomes in \(A\) are also in \(B\), then it is called a subset of \(B\) and can be written as \(A\subset B\). In the diagram below \(A\) is a subset of \(B\).

Example

Consider an experiment where the sample space includes all possible outcomes of rolling two dice at the same time. The sample space \((S)\) can be shown as:

$$ S=\left\{ (1,1),(1,2),(1,3),(1,4),(1,5),(1,6),\\ (2,1),(2,2),(2,3),(2,4),(2,5),(2,6),\\ (3,1),(3,2),(3,3),(3,4),(3,5),(3,6),\\ (4,1),(4,2),(4,3),(4,4),(4,5),(4,6),\\ (5,1),(5,2),(5,3),(5,4),(5,5),(5,6),\\ (6,1),(6,2),(6,3),(6,4),(6,5),(6,6) \right\} $$

Define event \(E\) as “rolling doubles” and event \(F\) as “contains an odd number”. The outcomes of event \(E\) and event \(F\) can be shown as:

$$ E=\left\{ {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)} \right\} $$

$$ \begin{align*} F=\left\{ (1,1), (1,2), (1,3), (1,4), (1,5), (1,6),\\ (2,1), (2,3), (2,5), (3,1), (3,2), (3,3), \\ (3,4), (3,5), (3,6), (4,1), (4,3), (4,5), \\ (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), \\ (6,1), (6,3), (6,5) \right\} \end{align*} $$

The intersection of events \(E\) and \(F\) can be shown as:

$$ E\cap F=\left\{ { (1,1),(3,3),(5,5) } \right\} $$

The complement of event \(F\) can be written as:

$$ { F }^{ \prime }={ \left\{ (2,2),(2,4),(2,6),(4,2),(4,4),(4,6),(6,2),(6,4),(6,6) \right\} } $$

Probability

Given an experiment repeating \(n\) times, the probability of event \(A\) occurring can be calculated by dividing the number of times event \(A\) occurs by the number of times the experiment was performed, as shown in the formula below:

$$ P\left( A \right) =probability\quad of\quad event\quad A\quad occurring=\frac { \sharp \quad of\quad times\quad event\quad A\quad occurs }{ n } $$

where \(n\) is the number of times the experiment was performed.

And as \(n\) approaches infinity or, in other words, as the number of times the experiment is performed becomes infinitely large, this ratio approaches the true value of the probability of event \(A\) occurring.

Axioms of Probability

It is also useful to consider some general rules of probability, also called axioms of probability. The three axioms of probability are defined below.

Given \(S\) is sample space of an experiment and \(A\), \(B\) and \(E\) are events, then

  1. \(0 ≤= P(E) ≤=1\),meaning the probability of event \(E\) occurring is greater than or equal to 0 (or 0%) and less than or equal to 1 (or 100%).
  2. \(P(S) = 1\), meaning with a probability of 1 (or 100%), the outcome of an experiment will be in the sample space, \(S\).
  3. \(P\left( A\cup B \right) =P(A)+P(B)\) if \(A\) and \(B\) are mutually exclusive, meaning the probability of event \(A\) or event \(B\) occurring is the sum of the probability of event \(A\) and the probability of event \(B\).

Example

Given the experiment above involving rolling two dice simultaneously, what is the probability of rolling doubles?

 

If we define the event, \(E\), as “rolling doubles” then

$$ E=\left\{ { (1,1),(2,2),(3,3),(4,4),(5,5),(6,6) }, \right\} \quad and $$

$$ P\left( E \right) =\frac { \sharp \quad of\quad times\quad event\quad E\quad occurs }{ n } =\frac { 6 }{ 36 } ={ 1 }/{ 6 } $$

 

Learning Outcome

1.a Topic: General Probability – Define set functions, Venn diagrams, sample space, and events. Define probability as a set function on a collection of events and state the basic axioms of probability. 


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