# 3.2 Probability of An Event

Given an event [latex]E[/latex], the chance or the probability that the event [latex]E[/latex] happens is denoted as [latex]P(E)[/latex]. A probability near 0 indicates that the event is very unlikely to occur when the chance experiment is conducted, whereas a probability near 1 suggests that the event is very likely to occur.

**3.2.1 Frequentist Interpretation of Probability**

From the frequentist’s point of view, the probability of an event can be interpreted as the proportion of times the event occurs in a large number of repetitions of the chance experiment. For instance, if we flip a balanced coin 100 times and observe 55 heads, then the probability of observing a head is

[latex]P(H) \approx \frac{\text{# of times we observe a head}}{n} = \frac{55}{100} = 0.55.[/latex]

The figure below shows that the proportion approaches to 0.5 as the number of experiments repetitions [latex]n[/latex] increases. It is expected to observe heads half of the time, because the coin is balanced and there is a 50–50 chance heads are observed. Therefore, the proportion of observed heads will approach a constant when the coin is flipped infinite times, this constant is [latex]P(H).[/latex]

Theoretically speaking, it is impossible to observe the probability of an event [latex]P(E)[/latex], because we won’t repeat the chance experiment infinite times. However, we can sometimes calculate the probability based on a model or some probability rules.

## 3.2.2 Equally Likely Outcome Model, the f/N Rule

The simplest model, the equally likely outcome model, assumes that all possible outcomes have equal chance to be observed, such as flipping a **balanced** coin and rolling a **fair **die. For the equally likely outcome model, the probability that an event E happens is given by

[latex]\begin{align*} P(E) &= \frac{\text{# of sample points in event }E}{\text{# of sample points in sample space }S} \\ &= \frac{\text{# of ways event E can occur}}{\text{# of possible outcomes}} \\ &= \frac{f}{N}. \end{align*}[/latex]

Example: Equally Likely Outcomes

- Recall that a standard die has sample space [latex]S[/latex]= {1, 2, 3, 4, 5, 6}. Use the equally likely outcomes model to find the probability of the following events:

- Observing a six: E = {6}
[latex]P(E) = \frac{\text{# of sample points in event E}}{\text{# of sample in sample space S}} = \frac{f}{N} = \frac{1}{6}.[/latex]

- Observing an even number: A = {2, 4, 6}

[latex]P(A) = \frac{\text{# of sample points in event A}}{\text{# of sample points in space S}} = \frac{f}{N} = \frac{3}{6}=\frac{1}{2}.[/latex]

- Observing an outcome that is less than 3: B = {1, 2}

[latex]P(B) = \frac{\text{# of sample points in event B}}{\text{# of sample points in space S}} = \frac{f}{N} = \frac{2}{6} = \frac{2}{3}.[/latex]

- Suppose there are 100 students in a class, the following table summarizes the frequencies of number of siblings the students have:

**Table 3.2**: Frequency and Relative Frequency of # of Siblings

Total | 100 | 1.00 |

# of Siblings |
Frequency |
Relative Frequency |

0 | 10 | 0.10 |

1 | 30 | 0.30 |

2 | 35 | 0.35 |

3 | 15 | 0.15 |

>3 | 10 | 0.10 |

Find the probability that a randomly selected student has:

- exactly one sibling

If we randomly pick one student, each student has the same chance to be chosen; therefore, we can use the equally likely outcome model. There are [latex] f=30[/latex] students with exactly one sibling. Hence

[latex]P(E) = \frac{\text{# of sample points in event E}}{\text{# of sample points in sample space S}} = \frac{f}{N} = \frac{30}{100} = 0.3.[/latex]

- at least one sibling, which means one, two, three, or more than three siblings.

[latex]P(E) = \frac{\text{# of sample points in event E}}{\text{# of sample points in sample space S}} = \frac{f}{N} = \frac{30+35+15+10}{100} = 0.9.[/latex]

- two to three siblings (inclusive), which means either two or three siblings.

[latex]P(E) = \frac{\text{# of sample points in event E}}{\text{# of sample points in sample space S}} = \frac{f}{N} = \frac{35+15}{100} = 0.5.[/latex]

Key Facts: Basic Properties of the Probability of an Event

- The probability of an event P(E) is always between 0 and 1, that is, [latex]0 \leq P(E) \leq 1[/latex].
- The probability of an event that can never occur is 0, e.g., P (observe a 7 when roll a regular die) = 0.
- The probability of an event that must occur is 1, e.g., P (observe a number smaller than 7 when roll a regular die) = 1.

All these properties can be easily shown by the f/N rule.

It is straightforward to show that using the f/N rule:

- [latex]P(\varnothing) = 0[/latex], where [latex]\varnothing[/latex] is the empty set, a set with no element.
- [latex]P(S) = 1[/latex]