2.5 - What is Probability (Formally)?

Previously, we defined probability informally. Now, let's take a look at a formal definition using the “axioms of probability.”

Probability of the Event

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:

  1. The probability of any event \(A\) must be nonnegative, that is, \(P(A)\ge 0\).
  2. The probability of the sample space is 1, that is, \(P(\mathbf{S})=1\).
  3. Given mutually exclusive events \(A_1, A_2, A_3, ...\) that is, where \(A_i\cap A_j=\emptyset\), for \(i\ne j\),

    \(P(A_1\cup A_2 \cup \cdots \cup A_k)=P(A_1)+P(A_2)+\cdots+P(A_k)\)

    \(P(A_1\cup A_2 \cup \cdots )=P(A_1)+P(A_2)+\cdots \)

    1. the probability of a finite union of the events is the sum of the probabilities of the individual events, that is:
    2. 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.