4: Probability
- Page ID
- 216187
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In this chapter, you will learn to:
- Write sample spaces.
- Determine whether two events are mutually exclusive.
- Use the Addition Rule.
- Calculate probabilities using both tree diagrams and combinations.
- Do problems involving conditional probability.
- Determine whether two events are independent.
- 4.1: Sample Spaces and Probability
- This page explores the fundamentals of probability, focusing on sample spaces and their roles in determining outcomes of events, such as rolling dice and drawing marbles. It clarifies common misconceptions about probabilities and defines probability's range and relationship to events. The text discusses different sampling methods, specifically with and without replacement, and explains how these affect sample spaces and probabilities.
- 4.2: Mutually Exclusive Events and the Addition Rule
- This page covers compound events in probability, explaining union, intersection, and complement concepts. Union (E ∪ F) is when either or both events occur, and intersection (E ∩ F) is when both happen simultaneously. It introduces the Addition Rule for calculating P(E ∪ F) to avoid double counting, and the Complement Rule, noting that the probability of an event not occurring is 1 - P(E). Examples include card draws and outcomes from rolling dice, illustrating these principles effectively.
- 4.3: Probability Using Tree Diagrams and Combinations
- This page explains probability calculations using tree diagrams and combinations, focusing on independent and conditional probabilities with examples, such as drawing marbles and card pairs. It highlights the significance of combinations in selecting items and includes practical applications, like the Birthday Problem and configurations of cell phone types.
- 4.4: Conditional Probability
- This page introduces and elaborates on conditional probability, defining it and demonstrating its significance through various examples. It explains the concept of \(\mathrm{P(E | F)}\) and provides foundational formulas for calculating conditional probabilities. Examples include rolling dice, analyzing transportation preferences, and student subscriptions to streaming services like Amazon Prime and Netflix, illustrating how event interrelations affect probabilities.
- 4.5: Independent Events
- This page covers the concepts of independent and dependent events in probability, explaining how the occurrence of one event can affect another's probability. It defines independent events, provides examples (using cards and flights), and discusses methods to check independence. Additionally, it investigates the independence of three events \(E\), \(F\), and \(G\), revealing the independence relations among them, and introduces the multiplication rule for non-independent events.
- 4.6: Review of Probability
- This page features a problem set on calculating probabilities across multiple scenarios, including rolling dice, drawing cards, and selecting items. It discusses independent and mutually exclusive events, as well as conditional probabilities, through exercises involving family situations, academic courses, and consumer decisions. The problems aim to illustrate the probability of specific outcomes in various contexts, such as traffic conditions and group selections.