# 2.5 - What is Probability (Formally)?

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

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.

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