Chapter_4_Slides_Jaggia1

Chapter 4: Introduction to Probability

Overview

• The chapter covers fundamental concepts in probability necessary for business analysis.

• Key topics include:

  • Fundamental Probability Concepts

  • Rules of Probability Application

  • Calculating Probabilities from Contingency Tables

  • Total Probability Rule and Bayes’ Theorem

Uncertainties

• Managers analyze uncertainties in decision-making, such as:

  • Chances of sales decrease with price increase

  • Likelihood of productivity increase from new methods

  • Odds of investment profitability

Case Study: 24/7 Fitness Center

• A manager seeks a data-driven approach to contact new open house attendees.

• Data based on previous 400 attendees categorized by age group and enrollment outcome:

  • Example Outcomes:

    • Age: Between 30 and 50 → Not Enrolled

    • Age: Over 50 → Enrolled

Fundamental Probability Concepts

Definition of Probability

• Probability is a numerical measure of likelihood.

  • Scale: 0 (impossible) to 1 (certain):

    • 0 < P(Ei) < 1 for events

Statistical Experiments

• A statistical experiment is a process leading to different possible outcomes.

• Repeated experiments may yield different results, termed random experiments.

Sample Space

• The outcome space (S) consists of all possible outcomes of an experiment:

  • Examples:

    • Coin Toss: S = {Head, Tail}

    • Die Roll: S = {1, 2, 3, 4, 5, 6}

Multiple-Step Experiments

Counting Rule

• For multiple-step experiments, total outcomes can be calculated as:

  • Total Outcomes = n1 * n2 * ... * nk

• Tree diagrams can visually represent outcomes.

Example: KP&L

• Project phases include design and construction, with 3 possible completion times for each stage.

  • Design: n1 = 3 (2, 3, 4 months)

  • Construction: n2 = 3 (6, 7, 8 months)

• Total outcomes for project completion:

  • (3)*(3) = 9 outcomes

  • Adding a 3rd stage changes calculation to (2)(3)(3) = 18 outcomes.

Assigning Probabilities

Requirements

• Each experimental outcome probability must be between 0 and 1;

• The sum of all probabilities must equal 1.

Methods

1. Classical Method: Assumes equally likely outcomes (e.g., rolling a die).

2. Empirical Method: Based on observed data; needs a large number of repetitions for accuracy.

3. Subjective Method: Based on personal judgment or opinion.

Events and Their Probabilities

Definition

• An event is a collection of sample points and can be simple or composed of multiple outcomes.

  • Sample Points: Outcomes of an experiment;

  • Simple Event: A single outcome.

Types of Events

• Exhaustive Events: Groups containing all possible experimental outcomes.

• Mutually Exclusive Events: Cannot occur simultaneously; shared outcomes are null.

Calculating Probabilities

Addition Law

• To calculate the probability of A or B occurring:

  • P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Example: KP&L Projects

• Calculate the probability of various event outcomes based on completion data.

Conditional Probability

Definition

• The probability of an event based on the occurrence of another:

  • P(A|B) = P(A ∩ B) / P(B)

• Example: Finding a job increases from 0.80 to 0.90 with previous experience.

Multiplication Law

• For intersection probabilities:

  • P(A ∩ B) = P(A) * P(B|A)

• Example: Household newspaper subscriptions based on existing subscriptions.

Contingency Tables

Usefulness

• Helps to analyze relationships between two categorical variables.

• Displays frequencies of occurrences and helps calculate empirical probabilities.

  • Example: Promotion status of police officers categorized by gender.

Example Calculation

• Joint probability calculations based on contingency data for promoted and non-promoted individuals.

End of Chapter Summary

• Probability is an essential tool in business analytics for managing uncertainties and making informed decisions.

• Understanding how to calculate and interpret probabilities helps in assessing risks and outcomes in various business scenarios.