📈Intro to Probability for Business Unit 3 – Probability Basics: Concepts and Rules

Key Concepts and Terminology

• Probability quantifies the likelihood of an event occurring and ranges from 0 (impossible) to 1 (certain)
• An experiment is a process that generates well-defined outcomes
• An outcome is the result of a single trial of an experiment
• An event is a set of outcomes of an experiment that share a common property
• The sample space is the set of all possible outcomes of an experiment
• Mutually exclusive events cannot occur at the same time in a single trial (rolling a 1 and a 2 on a die)
• Exhaustive events encompass all possible outcomes in the sample space
• Independent events do not influence each other's probability (flipping a coin and rolling a die)

Sample Spaces and Events

• A sample space can be represented using set notation, where each element is a possible outcome
• Events are subsets of the sample space and can be represented using set notation
• The complement of an event A, denoted as A', includes all outcomes in the sample space that are not in A
• The union of two events A and B, denoted as A ∪ B, includes all outcomes that are in either A or B, or both
• The intersection of two events A and B, denoted as A ∩ B, includes all outcomes that are in both A and B
• The empty set, denoted as ∅, is an event that contains no outcomes and has a probability of 0
• The universal set, denoted as U or S, is the sample space itself and has a probability of 1

Probability Rules and Axioms

• The probability of an event A, denoted as P(A), is a number between 0 and 1, inclusive
• The probability of the sample space, P(S), is equal to 1
• The probability of the empty set, P(∅), is equal to 0
• The sum of the probabilities of all outcomes in the sample space is equal to 1
• For any two mutually exclusive events A and B, P(A ∪ B) = P(A) + P(B)
• For any two events A and B, P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
• The complement rule states that for any event A, P(A') = 1 - P(A)

Types of Probability

• Classical probability is used when all outcomes in the sample space are equally likely (rolling a fair die)
  • Calculated as the number of favorable outcomes divided by the total number of possible outcomes
• Empirical probability is based on observed data or experimental results (survey responses)
  • Calculated as the number of times an event occurs divided by the total number of trials
• Subjective probability is based on personal belief or judgment (estimating the likelihood of a project's success)
• Axiomatic probability is based on the probability axioms and is used in theoretical probability
• Geometric probability involves calculating probabilities based on geometric properties (dartboard problem)

Conditional Probability

• Conditional probability is the probability of an event A occurring given that another event B has already occurred, denoted as P(A|B)
• Calculated as P(A|B) = P(A ∩ B) / P(B), where P(B) > 0
• The multiplication rule states that for any two events A and B, P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B)
• Bayes' theorem allows for updating probabilities based on new information
  • P(A|B) = P(B|A) × P(A) / P(B), where P(B) > 0
• The law of total probability states that for a partition of the sample space {B1, B2, ..., Bn}, P(A) = P(A|B1) × P(B1) + P(A|B2) × P(B2) + ... + P(A|Bn) × P(Bn)

Independence and Dependence

• Two events A and B are independent if the occurrence of one does not affect the probability of the other
  • Mathematically, P(A|B) = P(A) and P(B|A) = P(B)
• For independent events A and B, P(A ∩ B) = P(A) × P(B)
• Dependent events influence each other's probabilities (drawing cards without replacement)
• For dependent events A and B, P(A ∩ B) ≠ P(A) × P(B)
• Conditional probability is used to calculate probabilities for dependent events
• Independence and dependence are important concepts in probability theory and have significant implications in various fields (finance, insurance, and machine learning)

Probability Trees and Diagrams

• Probability trees are graphical representations of a sequence of events and their associated probabilities
• Each branch of the tree represents a possible outcome, and the probability of that outcome is written along the branch
• The probability of a specific path is calculated by multiplying the probabilities along the branches in that path
• Probability trees are useful for visualizing and calculating probabilities for multi-stage experiments (multiple coin flips)
• Venn diagrams are used to represent relationships between events and their probabilities
  • Each event is represented by a circle, and the overlapping regions represent the intersections of events
• Venn diagrams help visualize concepts such as union, intersection, and complement of events
• Tree diagrams and Venn diagrams are essential tools for understanding and solving probability problems

Applications in Business Decision-Making

• Probability is used in various aspects of business decision-making, such as risk assessment, investment analysis, and marketing strategies
• In finance, probability is used to model stock prices, calculate expected returns, and assess portfolio risk (Value at Risk)
• Insurance companies use probability to determine premiums based on the likelihood of claims (actuarial science)
• In project management, probability is used to estimate the likelihood of completing a project within a given timeframe and budget
• Marketing teams use probability to analyze customer behavior, segment markets, and optimize pricing strategies
• Operations managers use probability to model supply chain uncertainties, inventory levels, and production processes
• Decision trees combine probability trees with decision nodes to analyze complex business decisions under uncertainty
• Sensitivity analysis is used to assess how changes in probabilities affect the expected outcomes of a decision