Chapter 3: Probability – Basic Concepts and Counting Principles

Probability: Basic Concepts and Counting

Introduction to Probability

Probability is a fundamental concept in statistics that quantifies the likelihood of events occurring in random experiments. It provides a mathematical framework for making predictions and informed decisions under uncertainty.

Key Terms in Probability

• Probability Experiment: An action or process that leads to well-defined results called outcomes.

• Outcome: The result of a single trial of a probability experiment.

• Sample Space (S): The set of all possible outcomes of a probability experiment.

• Event (E): A subset of the sample space; it may consist of one or more outcomes.

• Simple Event: An event that consists of a single outcome.

Examples: Sample Space and Events

• Example 1: Flipping a fair coin. Sample space: S = {Head, Tail}. Event E = "get a head" = {Head}.

• Example 2: Rolling a six-sided die. Event A = rolling a 4 (simple event); Event B = rolling at least a 4 (not simple, as it includes 4, 5, and 6).

Tree Diagrams and Sample Spaces

Tree diagrams are useful for visualizing all possible outcomes of multi-stage experiments. Each branch represents a possible outcome at each stage.

• Example: Determining blood type (O, A, B, AB) and Rh factor (+ or -) yields 8 possible outcomes.

The Fundamental Counting Principle

The Fundamental Counting Principle states that if one event can occur in m ways and a second event in n ways, then the two events together can occur in m × n ways. This principle extends to any number of sequential events.

• Formula: For k events with n1, n2, ..., nk possible outcomes each, the total number of outcomes is $n_1 \times n_2 \times \cdots \times n_k$.

• Example: Choosing a car (3 manufacturers, 2 sizes, 4 colors): $3 \times 2 \times 4 = 24$ ways.

Counting Principle in Access Codes

• Example: Four-digit access code, digits 0-9.

• Digits not repeated: $10 \times 9 \times 8 \times 7 = 5040$ codes.

• Digits can be repeated: $10^4 = 10,000$ codes.

• First digit not 0 or 1, digits can repeat: $8 \times 10 \times 10 \times 10 = 8000$ codes.

Types of Probability

Classical (Theoretical) Probability

Classical probability applies when all outcomes in the sample space are equally likely.

• Formula: $P(E) = \frac{N(E)}{N(S)}$

• Example: Rolling a die, probability of rolling a 3: $P(3) = \frac{1}{6}$

Empirical (Statistical) Probability

Empirical probability is based on observed data from experiments or surveys.

• Formula: $P(E) = \frac{f}{n}$, where f is the frequency of event E, and n is the total number of trials.

• Example: In a survey of 1490 adults, 578 read only print books. $P(\text{read only print books}) = \frac{578}{1490} \approx 0.388$

Law of Large Numbers

The Law of Large Numbers states that as an experiment is repeated many times, the empirical probability approaches the theoretical probability.

Subjective Probability

Subjective probability is based on intuition, educated guesses, or estimates rather than precise calculations or data.

• Example: A doctor estimates a 90% chance of recovery for a patient.

Classifying Probability Statements

• Classical: Probability of winning a 1000-ticket raffle with one ticket: $\frac{1}{1000}$

• Empirical: Probability from survey data (e.g., 0.3 probability a voter is under 35).

• Subjective: Probability of getting an A on a test is 0.9 (based on personal judgment).

Range of Probabilities Rule

The probability of any event E is always between 0 and 1, inclusive:

• $0 \leq P(E) \leq 1$

Complementary Events

The complement of event E, denoted E', consists of all outcomes in the sample space not in E. The probabilities of an event and its complement always sum to 1.

• Formulas:

  • $P(E) + P(E') = 1$

  • $P(E) = 1 - P(E')$

  • $P(E') = 1 - P(E)$

Example: Probability of the Complement

• Probability that a randomly selected adult read no book: $P(\text{Read no book}) = \frac{395}{1490} = 0.265$

• Probability that a randomly selected adult read a book: $P(\text{Read book}) = 1 - 0.265 = 0.735$

Tree Diagrams and Probability Calculations

Using Tree Diagrams

Tree diagrams help visualize all possible outcomes in multi-stage experiments and are useful for calculating probabilities of compound events.

• Example: Tossing a coin and spinning a spinner with 8 numbers.

• Event A: Tossing a tail and spinning an odd number. There are 4 such outcomes out of 16 total: $P = \frac{4}{16} = 0.25$

• Event B: Tossing a head or spinning a number greater than 3. There are 13 such outcomes: $P = \frac{13}{16} \approx 0.813$

Application: Probability of a Specific Identification Number

• College ID number: 8 digits, each 0-9, digits can repeat.

• Total possible numbers: $10^8 = 100,000,000$

• Probability of randomly generating your specific ID: $P = \frac{1}{100,000,000} = 0.00000001$