Chapter 3: Probability – Structured Study Notes for Statistics Students

Probability

Basic Concepts of Probability and Counting

Probability is a fundamental concept in statistics that quantifies 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: Determining the sample space for blood types (O, A, B, AB) and Rh factor (positive or negative) yields eight possible outcomes.

Simple Events

A simple event consists of a single outcome, while a compound event includes more than one outcome. Identifying simple events is crucial for calculating probabilities accurately.

• Simple Event: An event with only one outcome (e.g., selecting a specific defective part).

• Compound Event: An event with multiple outcomes (e.g., rolling at least a 4 on a die).

Example: Rolling a 4, 5, or 6 on a die is not a simple event because it includes three outcomes.

The Fundamental Counting Principle

The Fundamental Counting Principle is used to 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: For k events, each with n_i possible outcomes: $n_1 \times n_2 \times \ldots \times n_k$

Example: Choosing a manufacturer, car size, and color from the table below:

Manufacturer	Car size	Color
Ford	compact, midsize	white (W), red (R), black (B), green (G)
GM	compact, midsize	white (W), red (R), black (B), green (G)
Honda	compact, midsize	white (W), red (R), black (B), green (G)

There are 3 manufacturers × 2 sizes × 4 colors = 24 possible combinations.

Counting with Restrictions

When restrictions are applied, such as digits not being repeated or certain digits excluded, the Fundamental Counting Principle adapts accordingly.

• Example: Access code with four digits (0–9):

• Each digit can be used only once: $10 \times 9 \times 8 \times 7 = 5040$

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

• Each digit 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 is equally likely. $P(E) = \frac{n(E)}{n(S)}$

• Empirical (Statistical) Probability: Based on observed data. $P(E) = \frac{\text{frequency of event}}{\text{total frequency}}$

• Subjective Probability: Based on intuition, educated guesses, or estimates.

Example: Rolling a die:

• Event A: Rolling a 3. $P(A) = \frac{1}{6}$

• Event B: Rolling a 7. $P(B) = 0$ (not possible)

• Event C: Rolling less than 5. $P(C) = \frac{4}{6} = \frac{2}{3}$

Empirical Probability in Practice

Empirical probability uses real-world data to estimate the likelihood of events.

• Example: Survey of U.S. adults on book reading:

• Probability that the next adult surveyed read only print books: $P = \frac{560}{1502} \approx 0.373$

Example: Probability of a social media user being 25–34 years old:

Ages	Frequency, f
13 to 17	84
18 to 24	459
25 to 34	765
35 to 44	546
45 to 54	432
55 to 64	369
65 and over	345
Total	3000

• Probability: $P = \frac{765}{3000} = 0.255$

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 personal judgment, intuition, or experience rather than formal calculations.

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

Range of Probabilities Rule

The probability of any event is always between 0 and 1, inclusive. This is known as the range of probabilities rule.

• Formula: $0 \leq P(E) \leq 1$

Complementary Events

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

• Formula: $P(E) + P(E') = 1$

• $P(E) = 1 - P(E')$

• $P(E') = 1 - P(E)$

Tree Diagrams in Probability

Tree diagrams are visual tools used to enumerate all possible outcomes of a probability experiment, especially when multiple events occur in sequence.

• Example: Tossing a coin and spinning a spinner with numbers 1–8.

• 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).

Application of the Fundamental Counting Principle

For identification numbers or codes, the Fundamental Counting Principle helps determine the total number of possible combinations.

• Example: College ID number with eight digits (0–9, digits can repeat): $10^8 = 100,000,000$ possible numbers.

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

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