Chapter 4: Probability – Introductory Statistics Study Notes

Probability

Introduction to Probability

Probability is a fundamental concept in statistics that quantifies how likely an event is to occur. It provides a mathematical framework for making predictions and decisions based on data.

• Probability: How often something occurs or is expected to occur.

• Probability gives us information about data in a widely accessible format.

• Probability theory can be used to make decisions about populations, such as by applying the Rare Event Rule.

Rare Event Rule for Inferential Statistics

The Rare Event Rule is used to determine whether an observed event is so unlikely under a given assumption that the assumption should be questioned.

• If, under a given assumption, the probability of an observed event is extremely small, then the assumption is probably not correct.

• Such an event is called statistically significant because it is too unlikely to occur just by chance.

• This rule is foundational for hypothesis testing and will be revisited later in the course.

Basic Probability Terms

Key Definitions

Understanding basic probability terms is essential for analyzing random processes and outcomes.

• Event: An outcome of a procedure.

• Simple event: An event that cannot be broken down into simpler components.

• Outcome: The result of a single trial in a procedure.

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

Sample Space and Tree Diagrams

Tree Diagrams

Tree diagrams are visual tools used to systematically list all possible outcomes (sample space) of a random experiment.

• Each branch represents a possible outcome at each stage.

• Tree diagrams help in finding probabilities and organizing complex experiments.

Examples of Sample Spaces

Sample spaces can be constructed for simple experiments such as rolling dice or flipping coins.

• One roll of a six-sided die: {1, 2, 3, 4, 5, 6}

• One coin flip: {Heads, Tails}

• Two coin flips: {HH, HT, TH, TT}

• Three coin flips: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

Complex Sample Spaces

Tree diagrams can be used for more complex experiments, such as games involving multiple steps.

• Example: Flip a coin, spin a spinner with red, green, and blue areas, then flip another coin.

• To find the total number of possible outcomes, multiply the number of outcomes for each step.

Fundamental Counting Principle

Definition and Application

The Fundamental Counting Principle is a method for determining the total number of possible outcomes in a multi-step experiment.

• Multiply the number of possible outcomes for each part of a procedure.

• Useful for calculating probabilities in complex scenarios.

Example: Car Selection

Suppose you are purchasing a new car. The possible manufacturers, car sizes, and colors are listed:

• Manufacturer: Ford, GM, Honda

• Car size: compact, midsize

• Color: white (W), red (R), black (B), green (G)

To find the total number of ways to select one manufacturer, one car size, and one color, multiply the number of choices for each category: $3 \times 2 \times 4 = 24$ ways.

Counting Examples

Counting principles are applied to various real-world scenarios:

• How many outfits can you make with 4 shirts, 3 pairs of pants, and 3 pairs of shoes? $4 \times 3 \times 3 = 36$ outfits.

• Internet password with five capital letters (no repeats): $26 \times 25 \times 24 \times 23 \times 22$ possible passwords.

• Door passcode with four digits, second digit must be four, last digit must be even: Apply restrictions and multiply possibilities.

• Ways to put 6 books on a shelf: $6!$ (factorial).

Factorial

Definition and Calculation

The factorial symbol, "!", denotes the product of decreasing positive whole numbers. Factorials are used in counting and probability calculations.

• For example, $4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24$

• By special definition, $0! = 1$

Factorial Examples

• Find the value of five factorial: $5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$

• Find the value of twenty-one factorial: $21!$ (very large number, typically calculated with a calculator or software)

Probability Rules

Basic Rules

Probability values are always between 0 and 1, inclusive. These rules help interpret and calculate probabilities.

• Probability is usually expressed as a fraction or decimal.

• For any event, $0 \leq P(A) \leq 1$

• If $P(A) = 0$, the event is impossible.

• If $P(A) = 1$, the event is certain.

Example: Valid Probability Values

Determine which numbers cannot be probabilities and explain why:

• Valid probabilities: 0, 1, 0.3948, 0.99, 0.0001

• Invalid probabilities: 15, 1.11, $\sqrt{5}$, -0.2938 (because they are not between 0 and 1)

Classical vs. Relative Frequency Probability

Definitions and Comparison

There are two main approaches to calculating probability: classical and relative frequency.

• Classical Probability: Each event has an equal chance of occurring; probability is calculated without actually producing the results.

• Relative Frequency Probability: Probability is calculated based on data from actual experiments; it is an approximation.

Calculating Probability

Probability Formula

The probability of an event A is calculated as the ratio of the number of occurrences of A to the total number of possible outcomes.

• Formula: $P(A) = \frac{\text{number of occurrences of A}}{\text{total number of outcomes}}$

• This formula applies to both classical and relative frequency probability.

Summary Table: Probability Concepts

Concept	Definition	Example
Event	Outcome of a procedure	Rolling a 4 on a die
Sample Space	All possible outcomes	{1,2,3,4,5,6} for a die
Classical Probability	Equal chance, theoretical	Probability of heads on a coin: 0.5
Relative Frequency Probability	Based on experiment	Observed heads in 100 flips
Factorial	Product of decreasing numbers	4! = 24
Probability Rule	0 ≤ P(A) ≤ 1	P(A) = 0.99