# 3.1 Basic Concepts in Probability

Let us first introduce some basic concepts in probability theory.

**Chance experiment**is a process producing outcomes that vary randomly when repeated.**Sample space,**denoted as [latex]S[/latex], is the collection of ALL possible outcomes of a chance experiment.- Each possible outcome in the sample space is called a
**sample point**. **An event**is a combination of sample points; it is a subset of the sample space. We use capital letters A, B, C, …, E, … to represent events.

Example: Basic Concepts

**Table 3.1**: Examples of Sample Space and Events

Chance Experiment |
Sample Space S |
Events |

Flip a balanced coin | {H, T} where H: head, T: tail | E = observe a head = {H} |

Roll a fair die | {1, 2, 3, 4, 5, 6} | E = observe a six = {6} A = observe even numbers = {2, 4, 6} B=outcome is less than 3 = {1, 2} |

Flip a balanced coin twice | {HH, HT, TT, TH} | E = observe the same outcome = {HH, TT} A = observe at least one head = {HH, HT, TH} |

For example, consider rolling a fair die. The sample space [latex]S = \{1, 2, 3, 4, 5, 6\}[/latex] consists of six sample points, while the event observing even numbers [latex]A = \{2, 4, 6 \}[/latex]** **contains three sample points, which are part of the sample space.

Exercise: Basic Concepts

Consider the chance experiment of rolling a fair die twice.

- Identify the sample space
*S*. - List all possible outcomes of the event that the two rolls give the same result.
- List all possible outcomes of the event that at least one six is observed.

## Show/Hide Answer

- The sample space [latex]S[/latex] contains [latex]6 \times 6[/latex] pairs in the form of (1, 1), (1, 2), …, (1, 6), (2, 1), (2, 2), …, (2, 6), …, (6,1), (6, 2), …, (6, 6).
- E = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}
- E = {(1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5)}