Previously, we defined probability informally. Now, let's take a look at a formal definition using the “axioms of probability.”
- Probability of the Event
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Probability is a (real-valued) set function \(P\) that assigns to each event \(A\) in the sample space \(\mathbf{S}\) a number \(P(A)\), called the probability of the event \(A\), such that the following hold:
- The probability of any event \(A\) must be nonnegative, that is, \(P(A)\ge 0\).
- The probability of the sample space is 1, that is, \(P(\mathbf{S})=1\).
- Given mutually exclusive events \(A_1, A_2, A_3, ...\) that is, where \(A_i\cap A_j=\emptyset\), for \(i\ne j\),
\(P(A_1\cup A_2 \cup \cdots \cup A_k)=P(A_1)+P(A_2)+\cdots+P(A_k)\)
\(P(A_1\cup A_2 \cup \cdots )=P(A_1)+P(A_2)+\cdots \)
- the probability of a finite union of the events is the sum of the probabilities of the individual events, that is:
- the probability of a countably infinite union of the events is the sum of the probabilities of the individual events, that is:
Example 2-8 Section
Suppose that a Stat 414 class contains 43 students, such that 1 is a Freshman, 4 are Sophomores, 20 are Juniors, 9 are Seniors, and 9 are Graduate students:
| Status | Fresh | Soph | Jun | Sen | Grad | Total |
| Count | 1 | 4 | 20 | 9 | 9 | 43 |
| Proportion | 0.02 | 0.09 | 0.47 | 0.21 | 0.21 |
Randomly select one student from the Stat 414 class. Defining the following events:
- Fr = the event that a Freshman is selected
- So = the event that a Sophomore is selected
- Ju = the event that a Junior is selected
- Se = the event that a Senior is selected
- Gr = the event that a Graduate student is selected
The sample space is S = (Fr, So, Ju, Se, Gr}. Using the relative frequency approach to assigning probability to the events:
- P(Fr) = 0.02
- P(So) = 0.09
- P(Ju) = 0.47
- P(Se) = 0.21
- P(Gr) = 0.21
Let's check to make sure that each of the three axioms of probability are satisfied.