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.