PSY 201

Course Information

• Instructor: Prof. Keisuke Fukuda

  • Office Hour: Mondays 11-12 @ CCT4067

  • Course: PSY201: Introduction to Quantitative Research in Psychology

• Lectures: Mondays 9-11

• Tutorials: Tuesdays and Wednesdays

Probability: Understanding Uncertainty

Definitions

• Probability: The branch of mathematics that deals with likelihood and uncertainty.

• Sample Space: The collection of all possible outcomes of a chance experiment.

  • Examples:

  • Single Die Roll: Sample space = {1, 2, 3, 4, 5, 6}

  • Two Dice Roll: Sample space = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

  • Customer Purchase: Male or female purchasing a Honda Civic (hybrid or traditional) as sample space = {(male, hybrid), (male, traditional), (female, hybrid), (female, traditional)}

• Chance Experiment: An activity or situation in which there is uncertainty about which of two or more possible outcomes will result.

Events

• Event: Any collection of outcomes of a chance experiment.

• Simple Event: An event consisting of exactly one outcome.

  • Examples of Simple Events:

  • Male purchasing a hybrid Civic (MH)

  • Female purchasing a traditional Civic (FT)

  • Examples of Events:

  • A customer purchasing a hybrid Civic: {MH, FH}

  • Female purchasing any kind of Civic: {FH, FT}

Creating and Defining Events

• Let A and B denote two events:

  • Not A: Event of all outcomes not in event A; also denoted as $A^{c}$, $A'$, or $¬A$.

  • A or B: Outcomes in at least one of A or B; called the union of the two events, denoted as $A igcup B$.

  • A and B: Outcomes in both A and B; called the intersection of the two events, denoted as $A igcap B$.

Venn Diagrams

• Definition: A Venn diagram is a graphical representation of events.

• Mutually Exclusive Events:

  • When A and B are mutually exclusive (disjoint), they cannot happen simultaneously.

Probability Calculations

• Probability of an Event (P(E)): The likelihood that event E occurs, where $0 \leq P(E) \leq 1$.

• Estimating Probability:

  • When many chance experiments are performed, $P(E)$ can be estimated as relative frequency:

  • For example, for a coin toss:

  • Relative Frequency of Heads can be calculated over a series of tosses.

• Fundamental Rules of Probability:

  1. P(A) = Probability of event A occurring

  2. P(not A) = Probability of event A not occurring

  • Relationship: $P(A) + P(not A) = 1$

  1. For two events A and B:

  • If A and B are mutually exclusive, $P(A \bigcup B) = P(A) + P(B)$

  • If A and B are not mutually exclusive, $P(A \bigcup B) = P(A) + P(B) - P(A \bigcap B)$

Conditional Probability

• P(A|B): The probability of event A occurring given that event B occurred.

  • Relationship: $P(A|B) = \frac{P(A \bigcap B)}{P(B)}$

Independence of Events

• When A and B are independent: The occurrence of A provides no information on the occurrence of B.

  • Relationship:

  • $P(A | B) = P(A)$ and $P(B | A) = P(B)$

  • $P(A \bigcap B) = P(A) \cdot P(B)$

Examples and Applications

• Imaginary Car Dealership Example:

  • Probability of Selling Japanese Cars:

  • Probability of selling Honda = 0.25

  • Probability of selling Nissan = 0.18

  • Probability of selling Toyota = 0.14

• Imaginary University Example:

  • To calculate the likelihood of a friend's living arrangement given they are a senior:

  • $P(On-campus|Senior)$ calculated from relevant data.

• Lyme Disease Detection Example:

  • Symptoms and blood test results presented in context of probabilities.

• Bayes' Rule:

  • For conditional probabilities where the events involved are not mutually exclusive, the relationship can be structured using Bayes' Theorem.

Bayes' Rule

• General Form: $P(A|E) = \frac{P(E|A) \cdot P(A)}{P(E|A) \cdot P(A) + P(E|B) \cdot P(B)}$

  • This rule can apply when multiple events are overlapping or influence each other significantly, especially noteworthy in medical testing or diagnostics.