Probability and Probability Distributions: Core Concepts for Introductory Statistics

Probability: Quantifying Uncertainty

Definition and Interpretation

• Probability is a numerical measure of how likely an event is to occur, denoted by P.

• Probability values range from 0 (impossible event) to 1 (certain event).

• Probabilities help quantify uncertainty and make predictions about random events.

Basic Probability Terms

• Experiment: An act or process that produces an outcome that cannot be predicted with certainty (e.g., rolling a die).

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

• Event: Any subset of the sample space.

Basic Probability Formula

• The probability of an event A is given by: $P(A) = \frac{\text{Number of outcomes in } A}{\text{Total number of outcomes in } S}$

• For any event A: $0 \leq P(A) \leq 1$

• For the sample space S: $P(S) = 1$

Example:

• Rolling a fair six-sided die: $S = \{1,2,3,4,5,6\}$

• Event B: rolling a 2 or 5, $B = \{2,5\}$

• $P(B) = \frac{2}{6} = \frac{1}{3}$

Operations with Events

Venn Diagrams

• Visual tools to represent relationships among events.

• Circles represent events; overlap shows shared outcomes; the rectangle is the sample space.

Intersection of Events

• Intersection (A ∩ B): Outcomes common to both events A and B.

• Notation: $A \cap B$

• Example: If $A = \{2,4,5\}$ and $B = \{3,5,6\}$, then $A \cap B = \{5\}$

• $P(A \cap B)$ is the probability both A and B occur.

Union of Events

• Union (A ∪ B): Outcomes in A, or B, or both.

• Notation: $A \cup B$

• Example: $A = \{2,4\}$, $B = \{1,5\}$, $A \cup B = \{1,2,4,5\}$

• $P(A \cup B)$ is the probability at least one of A or B occurs.

Disjoint (Mutually Exclusive) Events

• Events that cannot occur at the same time: $A \cap B = \varnothing$

• For disjoint events, $P(A \cap B) = 0$

Addition Rule of Probability

• General Addition Rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

• If A and B are disjoint: $P(A \cup B) = P(A) + P(B)$

Complement of an Event

• The complement of event A, denoted $A^c$ or $\overline{A}$, contains all outcomes in S not in A.

• $A \cup A^c = S$ and $A \cap A^c = \varnothing$

• Complement Rule: $P(A^c) = 1 - P(A)$

• For "at least one" problems: $P(\text{at least one}) = 1 - P(\text{none})$

Conditional Probability and Independence

Conditional Probability

• The probability that event A occurs given that event B has occurred.

• Notation: $P(A|B)$

• Formula: $P(A|B) = \frac{P(A \cap B)}{P(B)}$, provided $P(B) > 0$

• "Given that" signals conditional probability.

Multiplication Rule

• Probability that both A and B occur: $P(A \cap B) = P(A)P(B|A) = P(B)P(A|B)$

Independence

• Events A and B are independent if knowing one occurred does not change the probability of the other.

• Mathematically: $P(A|B) = P(A)$ or equivalently, $P(A \cap B) = P(A)P(B)$

Probability Tables and Applications

Contingency Tables

Contingency tables summarize data for two categorical variables. Probabilities can be calculated from the table by dividing the relevant cell count by the total.

	High Shock	Low Shock	Total
High Scratch	70	9	79
Low Scratch	16	5	21
Total	86	14	100

• Probability of high scratch resistance: $P(\text{High Scratch}) = \frac{79}{100} = 0.79$

• Probability of high scratch and high shock: $P(\text{High Scratch} \cap \text{High Shock}) = \frac{70}{100} = 0.70$

• Probability of high scratch but low shock: $P = \frac{9}{100} = 0.09$

Law of Total Probability

• When an event can occur through several pathways: $P(A) = P(B)P(A|B) + P(B^c)P(A|B^c)$

Random Variables and Probability Distributions

Random Variables

• A random variable assigns a numerical value to each outcome of a random experiment.

• Denoted by capital letters (e.g., X); possible values by lowercase (e.g., x).

Probability Distribution

• Describes how probabilities are assigned to all possible values of a random variable.

• Must satisfy:

  • For all x, $0 \leq P(x) \leq 1$

  • $\sum P(x) = 1$

Example: Coin Toss

X (Number of Heads)	P(X)
0	0.25
1	0.50
2	0.25

Expected Value (Mean) of a Random Variable

• The expected value (mean) is the long-run average outcome:

$E(X) = \mu = \sum x \cdot P(x)$

• Interpretation: Over many repetitions, the average value observed.

Example: Customer Service Calls

X (Calls)	0	1	2	3	4	5
P(X)	0.40	0.20	0.15	0.10	0.10	0.05

• Probability at least 3 calls: $P(X \geq 3) = 0.10 + 0.10 + 0.05 = 0.25$

• Mean number of calls: $E(X) = 0 \times 0.40 + 1 \times 0.20 + 2 \times 0.15 + 3 \times 0.10 + 4 \times 0.10 + 5 \times 0.05 = 1.55$

The Normal Distribution

Properties of the Normal Distribution

• Defined by mean $\mu$ and standard deviation $\sigma$.

• Symmetrical about the mean.

• Spread determined by $\sigma$.

• Notation: $N(\mu, \sigma)$

Empirical Rule (68–95–99.7 Rule)

• For normal distributions:

• ~68% of data within 1 standard deviation of the mean

• ~95% within 2 standard deviations

• ~99.7% within 3 standard deviations

Standard Normal Distribution

• Special case: mean 0, standard deviation 1.

• Random variable denoted by Z.

Finding Normal Probabilities

• To find $P(a \leq X \leq b)$, standardize X: $Z = \frac{X - \mu}{\sigma}$

• Use normal tables or software (e.g., StatCrunch) to find probabilities.

Inverse Normal Problems

• Given a probability, find the corresponding X-value (percentile).

• Use the inverse normal function in statistical software.

Summary Table: Key Probability Rules

Rule	Formula	When to Use
Addition Rule	$P(A \cup B) = P(A) + P(B) - P(A \cap B)$	Any two events
Addition Rule (Disjoint)	$P(A \cup B) = P(A) + P(B)$	Disjoint events
Multiplication Rule	$P(A \cap B) = P(A)P(B|A)$	Both events occur
Multiplication Rule (Independent)	$P(A \cap B) = P(A)P(B)$	Independent events
Complement Rule	$P(A^c) = 1 - P(A)$	Probability of "not A"
Conditional Probability	$P(A|B) = \frac{P(A \cap B)}{P(B)}$	Given B occurred

Learning Outcomes

• Define and interpret probability, events, and sample spaces.

• Use unions, intersections, and complements to describe events.

• Apply addition, multiplication, conditional, and complement rules to calculate probabilities.

• Distinguish between independent, dependent, and mutually exclusive events.

• Construct and interpret discrete probability distributions and calculate expected values.

• Calculate and interpret normal probabilities and percentiles using statistical software.

Additional info: Examples and applications were expanded for clarity. Some tables were reconstructed for completeness. All formulas are provided in LaTeX format as required.