Chapter 3: Basic Concepts of Probability and Counting (Elementary Statistics)

Chapter 3: Probability

Chapter Outline

• 3.1 Basic Concepts of Probability and Counting

• 3.2 Conditional Probability and the Multiplication Rule

• 3.3 The Addition Rule

• 3.4 Additional Topics in Probability and Counting

Section 3.1: Basic Concepts of Probability and Counting

Section Objectives

• Identify the sample space of a probability experiment and how to identify simple events

• Use the Fundamental Counting Principle to find the number of ways two or more events can occur

• Distinguish among classical probability, empirical probability, and subjective probability

• Find the probability of the complement of an event

• Use a tree diagram and the Fundamental Counting Principle to find probabilities

Probability Experiments

Probability is a foundational concept in statistics, used to quantify uncertainty and predict outcomes. The following definitions are essential:

• Probability experiment: An action or trial through which specific results (counts, measurements, or responses) are obtained.

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

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

• Event: Consists of one or more outcomes and is a subset of the sample space.

Identifying the Sample Space: Example

Consider a survey where people are asked for their blood type (O, A, B, AB) and Rh factor (positive or negative). The task is to determine the number of outcomes and identify the sample space.

• There are four blood types: O, A, B, AB.

• Each person is either Rh-positive (+) or Rh-negative (−).

• A tree diagram visually displays all possible outcomes by branching from a starting point.

Tree Diagram for Blood Types:

Blood Type	Rh Factor	Outcome
O	+	O+
O	−	O−
A	+	A+
A	−	A−
B	+	B+
B	−	B−
AB	+	AB+
AB	−	AB−

The sample space has eight possible outcomes: {O+, O−, A+, A−, B+, B−, AB+, AB−}

Simple Events

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

• Example: Tossing heads and rolling a 3 on a die is a simple event: {H3}

• Example: Tossing heads and rolling an even number is not a simple event: {H2, H4, H6}

The Fundamental Counting Principle

The Fundamental Counting Principle is used to determine the number of ways two or more events can occur in sequence.

• If one event can occur in $m$ ways and a second event can occur in $n$ ways, the number of ways the two events can occur in sequence is $m \times n$.

• This principle can be extended for any number of events occurring in sequence.

Example: If you have 3 manufacturers, 2 car sizes, and 4 colors, the number of possible car selections is:

• $3 \times 2 \times 4 = 24$ ways

Types of Probability

Probability can be classified into three main types:

• Classical (Theoretical) Probability: Each outcome in the sample space is equally likely. $P(E) = \frac{\text{Number of outcomes in event } E}{\text{Number of outcomes in sample space}}$

• Empirical (Statistical) Probability: Based on observations from experiments. $P(E) = \frac{\text{Frequency of event } E}{\text{Total frequency}}$

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

Range of Probabilities Rule

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

• $0 \leq P(E) \leq 1$

Complementary Events

The complement of event $E$ (denoted $E'$) is the set of all outcomes in the sample space that are not included in $E$.

• $P(E) + P(E') = 1$

• $P(E) = 1 - P(E')$

• $P(E') = 1 - P(E)$

Tree Diagrams

Tree diagrams are useful for visualizing all possible outcomes of a probability experiment, especially when events occur in sequence.

• Each branch represents a possible outcome at each stage.

• Tree diagrams help in counting outcomes and calculating probabilities.

Law of Large Numbers

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

Summary Table: Types of Probability

Type	Definition	Example
Classical	Equally likely outcomes	Rolling a die: $P(3) = \frac{1}{6}$
Empirical	Based on observed data	Survey: $P(\text{read only print books}) = \frac{560}{1502}$
Subjective	Based on intuition/guess	Doctor estimates 90% chance of recovery

Example Applications

• Blood Type Survey: Sample space for blood type and Rh factor is {O+, O−, A+, A−, B+, B−, AB+, AB−}.

• Car Selection: With 3 manufacturers, 2 sizes, and 4 colors, there are $3 \times 2 \times 4 = 24$ possible choices.

• Access Code: If each digit (0-9) can be used only once in a 4-digit code: $10 \times 9 \times 8 \times 7 = 5040$ codes.

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

• If first digit cannot be 0 or 1: $8 \times 10 \times 10 \times 10 = 8000$ codes.

Key Formulas

• Classical Probability: $P(E) = \frac{\text{Number of outcomes in } E}{\text{Number of outcomes in sample space}}$

• Empirical Probability: $P(E) = \frac{f}{n}$, where $f$ is frequency of event $E$, $n$ is total frequency

• Complement: $P(E') = 1 - P(E)$

Additional info:

• Tree diagrams and the Fundamental Counting Principle are essential tools for enumerating outcomes in multi-stage experiments.

• Understanding the types of probability helps in selecting the correct method for calculating probabilities in different contexts.