Lecture 5: Probability – Foundations and Event Operations

Probability: Concepts and Foundations

Overview of Probability

Probability is a fundamental concept in statistics, representing the measure or estimation of the likelihood that a particular event will occur. Probabilities are assigned values between 0 (impossible event) and 1 (certain event), and are used to quantify uncertainty in experiments and real-world situations.

• Probability: A numerical value expressing the chance of occurrence of an event.

• Range: 0 ≤ Probability ≤ 1.

• Application: Used in fields such as gaming, marketing, genetics, and more.

Quote: “The theory of probabilities is basically only common sense reduced to a calculus” – Pierre Laplace (1814)

Key Terminology

Understanding probability requires familiarity with several key terms:

• Experiment: Any action or process that generates observations (e.g., rolling a die).

• Outcome: The result of a single trial of an experiment (e.g., rolling a 4).

• Sample Space (S): The set of all possible outcomes of the experiment (e.g., S = {1, 2, 3, 4, 5, 6} for a fair die).

• Event: A collection of one or more outcomes (e.g., Event A = {2, 4, 6}, rolling an even number).

• Probability of Event E: Denoted as P(E), the likelihood that event E occurs.

Approaches to Assigning Probabilities

There are three main approaches to assigning probabilities to events:

• Classical Approach: Assumes all outcomes are equally likely. Probability of each outcome is $\frac{1}{k}$, where k is the number of possible outcomes.

• Subjective Approach: Probabilities are assigned based on expert opinion or personal judgment.

• Relative Frequency (Frequentist) Approach: Probability is estimated by performing n identical experiments and observing the frequency of the outcome.

Formula: $\text{Probability} = \frac{\text{Number of times outcome is observed}}{n}$

Examples

• Video Game Playability: 80 gamers test a game; 65 say it is playable. Probability that the game is playable: $\frac{65}{80} = 0.8125$

• Commuter Traffic Lights: Sample space for 3 intersections (stop or continue): S = {sss, ssc, scs, scc, css, csc, ccs, ccc}

• Birthday Month: Sample space is the set of 12 months.

Calculating Probability

Classical Probability Formula

When all outcomes are equally likely, the probability of an event E is:

$P(E) = \frac{\text{Number of ways E can occur}}{\text{Total number of possible outcomes}}$

• Example: Probability that a family with two children has at least one girl.

• Example: Probability of winning the lottery (matching 6 numbers from 47):

$\#S = \frac{47 \times 46 \times 45 \times 44 \times 43 \times 42}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 10,737,573$

$P(\text{Winning}) = \frac{1}{10,737,573}$

Event Operations

Complement of an Event

The complement of event A, denoted by A' or Ac, is the set of all outcomes not in A. The probability of the complement is:

$P(A') = 1 - P(A)$

• Interpretation: Probability that A does not occur.

Union of Events

The union of events A and B, denoted by A ∪ B, is the set of all outcomes that are in A or B (or both).

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

• Mutually Exclusive Events: If A and B are mutually exclusive (no overlap), $P(A \cup B) = P(A) + P(B)$

Intersection of Events

The intersection of events A and B, denoted by A ∩ B, is the set of all outcomes that are in both A and B.

• Mutually Exclusive Events: A ∩ B = ∅ (no outcomes in common).

Examples of Event Operations

• Blood Type Example: In Ireland, 55% have blood type O, 3% have AB. Probability of O or AB: $P(O \cup AB) = P(O) + P(AB) = 0.55 + 0.03 = 0.58$ (since O and AB are mutually exclusive).

• Marketing Campaign Example: 40% received email coupon, 30% saw social media ad, 15% saw both. Probability of at least one promotion: $P(\text{Email} \cup \text{Social Media}) = 0.40 + 0.30 - 0.15 = 0.55$

Summary of Probability Rules

• Probability is a measure of likelihood of occurrence of an event.

• Sample space (S): Set of all possible outcomes.

• Complement: $P(A') = 1 - P(A)$

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

• Mutually Exclusive: $P(A \cup B) = P(A) + P(B)$

Additional info: The next lecture will cover how to compute $P(A \cap B)$, which is essential for understanding dependent and independent events.