Chapter 3: Probability – Structured Study Notes for Statistics Students

Probability

Basic Concepts of Probability and Counting

Probability is a fundamental concept in statistics, describing the likelihood of events occurring in a random experiment. Understanding probability involves identifying the sample space, events, and outcomes, and applying counting principles to determine possible combinations.

• Probability Experiment: An action or trial that produces specific results (counts, measurements, or responses).

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

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

• Event: One or more outcomes; a subset of the sample space.

Example: Identifying the Sample Space

Suppose a survey asks for blood type (O, A, B, AB) and Rh factor (positive or negative). The sample space consists of all combinations:

• O+, O–, A+, A–, B+, B–, AB+, AB–

Simple Events

A simple event consists of a single outcome. If an event includes more than one outcome, it is not simple.

• Example: Selecting a specific defective machine part is a simple event (only one outcome).

• Example: Rolling at least a 4 on a die (outcomes: 4, 5, 6) is not a simple event.

The Fundamental Counting Principle

The Fundamental Counting Principle helps determine the number of ways multiple events can occur in sequence. If one event can occur in m ways and another in n ways, the total number of ways both can occur is m × n.

• Formula: $m \times n$

• Extended: For k events, $n_1 \times n_2 \times \ldots \times n_k$

Example: Car Selection

Suppose you can choose from 3 manufacturers, 2 car sizes, and 4 colors. The total number of combinations is:

• $3 \times 2 \times 4 = 24$ ways

Example: Access Codes

If a car's access code consists of four digits (0–9):

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

• Digits can be repeated: $10^4 = 10 \times 10 \times 10 \times 10 = 10,000$

• Digits can be repeated, but first digit cannot be 0 or 1: $8 \times 10 \times 10 \times 10 = 8,000$

Types of Probability

Probability can be classified into three main types: classical, empirical, and subjective.

• Classical (Theoretical) Probability: Each outcome in the sample space is equally likely. Formula: $P(E) = \frac{n(E)}{n(S)}$ Example: Rolling a 3 on a six-sided die: $P(3) = \frac{1}{6}$

• Empirical (Statistical) Probability: Based on observed data from experiments. Formula: $P(E) = \frac{\text{frequency of event}}{\text{total frequency}}$ Example: Probability that the next adult surveyed read only print books: $P = \frac{560}{1502}$

• Subjective Probability: Based on intuition, educated guesses, or estimates. Example: A doctor estimates a 90% chance of recovery.

Example: Using a Frequency Distribution

To find the probability that the next user surveyed is 25 to 34 years old:

• Frequency for 25–34: 765

• Total frequency: 3000

• $P = \frac{765}{3000}$

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.

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 not in E. The probabilities of E and E′ sum to 1:

• $P(E) + P(E') = 1$

• $P(E) = 1 - P(E')$

• $P(E') = 1 - P(E)$

Example: Probability of the Complement

Find the probability that a randomly selected social networking site user is not 25 to 34 years old:

• Complement: All users except those aged 25–34

• $P(E') = 1 - P(E)$

Tree Diagrams in Probability

Tree diagrams are visual tools used to display all possible outcomes of a probability experiment and to calculate probabilities for complex events.

Example: Coin and Spinner Experiment

• Event A: Tossing a tail and spinning an odd number (T1, T3, T5, T7)

• Event B: Tossing a head or spinning a number greater than 3 (H1–H8, T4–T8)

Additional Example: Fundamental Counting Principle

If your college ID number consists of eight digits (0–9, repeated):

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

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

Summary Table: Types of Probability

Type	Definition	Example
Classical	Equally likely outcomes	Rolling a die
Empirical	Based on observed data	Survey results
Subjective	Intuition or estimate	Doctor's prognosis

Key Formulas

• Classical Probability: $P(E) = \frac{n(E)}{n(S)}$

• Empirical Probability: $P(E) = \frac{\text{frequency of event}}{\text{total frequency}}$

• Complementary Events: $P(E') = 1 - P(E)$

• Fundamental Counting Principle: $n_1 \times n_2 \times \ldots \times n_k$

Conclusion

Probability is essential for understanding uncertainty and making informed decisions in statistics. Mastery of sample spaces, events, counting principles, and probability types provides a strong foundation for further study in statistics.