Probability: Foundations and Applications for Business Statistics

Probability of An Event

Experiment and Sample Space

An experiment is the process of observing a phenomenon that has variation in its outcomes. The set of all possible distinct outcomes of an experiment is called the sample space, denoted by $\Omega$.

• Elementary outcome: A single possible result of the experiment.

• Event: A set of elementary outcomes possessing a designated feature.

Types of Sample Space:

• Discrete sample space:

  • Finite: e.g., outcome of rolling a die, $\Omega = \{1,2,3,4,5,6\}$

  • Countable infinite: e.g., number of times to roll three sixes until observing 6-6-6, $\Omega = \{1,2,3,...\}$

• Continuous sample space:

  • e.g., the distance a car can travel with a full tank of gasoline, $\Omega = \{d : d \geq 0\}$

Definition: Probability of An Event

The probability of an event is a numerical value representing the proportion of times the event is expected to occur when the experiment is repeated many times under identical conditions.

• The probability of event $A$ is denoted by $P(A)$ or $Pr(A)$.

Example: Rolling a die, let event $A$ be 'even number'. $A = \{2,4,6\}$, so $P(A)$ is the probability of rolling an even number.

Axioms of Probability

Probability must satisfy the following axioms:

• $0 \leq P(A) \leq 1$ for all event $A$

• $P(\Omega) = 1$

• If $A_1, A_2, ...$ are mutually exclusive events, $P(\cup_i A_i) = \sum_i P(A_i)$

Note: These axioms apply to both discrete and continuous sample spaces.

Methods of Assigning Probability

Equally Likely Elementary Outcomes

If each elementary outcome is as likely to occur as any other, we use a uniform probability model. For an event $A$ consisting of $m$ elementary outcomes out of $k$ total outcomes:

• $P(A) = \frac{m}{k}$

Example: Rolling a die, event $A$ = 'even number' ($2,4,6$), $P(A) = 3/6 = 1/2$.

Probability as Long-Run Relative Frequency

Probability can also be defined as the value to which the relative frequency stabilizes with increasing number of trials.

• Although $P(A)$ is never known exactly, it can be estimated by repeating the experiment many times.

Example: Flipping a fair coin, the probability of observing a head is $1/2$.

Event Operations

Basic Event Relations

Events are subsets of the sample space. The three basic event relations are complement, union, and intersection.

• Complement: The set of all outcomes not in event $A$, denoted $A^c$ or $\bar{A}$.

• Union: The set of all outcomes in $A$, $B$, or both, denoted $A \cup B$.

• Intersection: The set of all outcomes in both $A$ and $B$, denoted $A \cap B$.

Venn Diagram

Venn diagrams visually represent event operations. For example, tossing a coin twice:

• Sample space: $\Omega = \{HH, HT, TH, TT\}$

• Event $A$: Tail at the second toss ($A = \{HT, TT\}$)

• Event $B$: At least one head ($B = \{HH, HT, TH\}$)

Event compositions:

• $\bar{A} = \{HH, TH\}$

• $\bar{B} = \{TT\}$

• $A \cup B = \{HT, TT, HH, TH\} = \Omega$

• $A \cap B = \{HT\}$

(Two) Laws of Probability

Law of Complement

• $P(A^c) = 1 - P(A)$

Addition Law

• $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Special Addition Law for Mutually Exclusive Events: If $A$ and $B$ are mutually exclusive, $P(A \cap B) = 0$, so $P(A \cup B) = P(A) + P(B)$.

Conditional Probability and Independence

Conditional Probability

The conditional probability of $A$ given $B$ is denoted $P(A|B)$ and defined as:

• $P(A|B) = \frac{P(A \cap B)}{P(B)}$

This is also called the multiplication law of probability:

• $P(A \cap B) = P(B) P(A|B)$

Example: The proportion of patients fearing visiting the dentist's office by education level:

	Elementary	Middle	High
Fear	0.12	0.08	0.05
No fear	0.28	0.25	0.22

What is the probability that a randomly selected patient is fearing visiting the dentist's office and is an elementary student?

• $P(\text{fear} \cap \text{elementary}) = 0.12$

• $P(\text{fear} | \text{elementary}) = \frac{0.12}{0.4} = 0.3$

Independence

Two events $A$ and $B$ are independent if:

• $P(A|B) = P(A)$

• Equivalently, $P(A \cap B) = P(A) P(B)$

The Power of the Multiplication Law

The multiplication law allows calculation of probabilities for compound events.

Example: In a class of 25 students (20 female, 5 male), the instructor randomly chooses 2 students. Calculate the probability that both are male:

• $P(M_1, M_2) = \frac{5}{25} \times \frac{4}{24} = \frac{1}{30} \approx 0.033$

Probability that one is male and one is female:

• $P(M_1, F_2) = \frac{5}{25} \times \frac{20}{24} = 1/6$

• $P(F_1, M_2) = \frac{20}{25} \times \frac{5}{24} = 1/6$

• Total probability = $1/6 + 1/6 = 1/3 \approx 0.333$

Additional info: The multiplication law simplifies calculations for sequential selections and is especially useful when events are independent or mutually exclusive.

Total Probability and Bayes' Rule

Total Probability

The law of total probability allows calculation of the probability of an event by partitioning the sample space:

• If $B_1, B_2, ..., B_n$ are mutually exclusive and exhaustive events, then for any event $A$:

• $P(A) = \sum_{i=1}^n P(A|B_i) P(B_i)$

Bayes' Rule

Bayes' rule allows updating probabilities based on new information:

• $P(B|A) = \frac{P(A|B) P(B)}{P(A)}$

Example: A manufacturer sources LED screens from three suppliers. Given probabilities for each supplier and defect rates, Bayes' rule can be used to find the probability that a defective screen came from a particular supplier.

• Let $B_i$ = supplier $i$, $A$ = defective screen.

• $P(B_i|A) = \frac{P(A|B_i) P(B_i)}{\sum_{j=1}^3 P(A|B_j) P(B_j)}$

Summary Table: Key Probability Laws

Law	Formula	Description
Complement	$P(A^c) = 1 - P(A)$	Probability of not $A$
Addition	$P(A \cup B) = P(A) + P(B) - P(A \cap B)$	Probability of $A$ or $B$
Multiplication	$P(A \cap B) = P(A) P(B|A)$	Probability of $A$ and $B$
Independence	$P(A \cap B) = P(A) P(B)$	If $A$ and $B$ are independent
Conditional	$P(A|B) = \frac{P(A \cap B)}{P(B)}$	Probability of $A$ given $B$
Total Probability	$P(A) = \sum_{i=1}^n P(A|B_i) P(B_i)$	Partitioned sample space
Bayes' Rule	$P(B|A) = \frac{P(A|B) P(B)}{P(A)}$	Update probability with new info

Additional info: These foundational concepts are essential for understanding probability in business statistics, including risk assessment, decision-making, and data analysis.