4.5: Vocabulary (Chapter 4)
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Chapter 4 Vocabulary
This list includes key terms introduced throughout Chapter 4. These terms form the foundation of probability, helping us describe, calculate, and interpret randomness in structured ways.
- Probability
- The long-run relative frequency of an event. Describes the proportion of times an event would occur if a random experiment were repeated many times under the same conditions. Values range from 0 to 1.
- Relative Frequency
- The proportion of times an event occurs out of the total number of trials in a simulation or experiment. Approximates probability in practice.
- Experiment
- An activity or process with an observable result. In probability, experiments are repeatable under identical conditions but have uncertain outcomes.
- Sample Space
- The set of all possible outcomes of a random experiment. Usually denoted by the symbol \( S \) or \( \Omega \).
- Event
- A set of outcomes (subset of the sample space) that satisfies a particular condition. Events can consist of one or more outcomes.
- Complement
- The set of all outcomes in the sample space that are not part of a given event. Denoted \( A^c \). Represented as "not A".
- Mutually Exclusive Events
- Two events that cannot occur at the same time. The intersection of mutually exclusive events is the empty set; \( P(A \cap B) = 0 \).
- Addition Rule
- A formula to find the probability that at least one of two events occurs: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] When A and B are mutually exclusive: \[ P(A \cup B) = P(A) + P(B) \]
- Conditional Probability
- The probability that an event A occurs, given that another event B has occurred. Notated as \( P(A \mid B) \) and calculated as: \[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \quad \text{provided } P(B) > 0 \]
- Independent Events
- Two events A and B are independent if knowing that one occurred does not affect the probability of the other. Mathematically: \[ P(A \cap B) = P(A) \cdot P(B) \quad \text{or equivalently} \quad P(A \mid B) = P(A) \]
- Dependent Events
- Events for which the outcome or occurrence of one affects the probability of the other. For dependent events, \( P(A \mid B) \neq P(A) \).
- Counting Rule (Multiplication Rule of Counting)
- If one task can be done in \( m \) ways and another in \( n \) ways, then the total number of ways to complete both tasks is \( m \times n \).
- Factorial
- The product of all positive integers less than or equal to a given number \( n \). Denoted \( n! \). For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).
- Permutation
- An arrangement of items where order matters. The number of ways to arrange \( r \) items from \( n \) is: \[ P(n, r) = \frac{n!}{(n - r)!} \]
- Combination
- A selection of items where order does not matter. The number of combinations of \( r \) items chosen from \( n \) is: \[ C(n, r) = \binom{n}{r} = \frac{n!}{r!(n - r)!} \]
Tip: Use these definitions as a reference while solving probability problems. Many of these ideas will return again when we explore inference, decision-making, and probability distributions later in the course.