4: Probability and Combinatorics
- Page ID
- 45472
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 4.1: Empirical Probability
- Empirical probability is based on observed outcomes from experiments rather than theoretical calculations. It uses the ratio of favorable outcomes to total trials, relying on the law of large numbers, which states that probabilities stabilize with more repetitions. Key terms include sample space (all possible outcomes) and event space (outcomes of interest).
- 4.2: Theoretical Probability
- Theoretical probability is based on known possible outcomes, assuming each is equally likely. Probabilities range from 0 to 1, where 0 means impossible and 1 means certain. Key rules include the complement rule, stating that the probability of an event not occurring is 1 minus the probability that it does occur.
- 4.3: Addition Rules for Probabilities
- The two addition rules for probability determine the chance of either one event or another occurring. If the events are mutually exclusive, meaning they cannot happen at the same time, you simply add their probabilities. If the events can occur together, you add their probabilities but subtract the overlap to avoid double-counting.
- 4.4: Multiplication Rules and Conditional Probability
- The multiplication rules help find the probability of two events happening together. If the events are independent, meaning one does not affect the other, you multiply their individual probabilities. For dependent events, where one event influences the other, conditional probability is used to adjust the calculation based on what has already occurred.
- 4.5: At Least One Rule and Tree Diagrams
- The "at least one" rule is used to find the probability that an event happens one or more times, often by first calculating the chance that it never happens. Tree diagrams are visual tools that map out all possible outcomes step-by-step, making it easier to organize and compute complex probabilities. Together, these methods help solve multi-step problems and avoid missed outcomes.
- 4.6: Counting Rules
- Counting rules help determine how many ways events can occur. The fundamental counting rule multiplies the number of choices for each step in a sequence. Permutations count arrangements where order matters, while combinations count selections where order does not matter.
- 4.7: Chapter 4 Formulas
- This section contains the key formulas for Chapter 4 material. It covers essential tools for solving probability problems, including the addition and multiplication rules, as well as counting techniques such as permutations and combinations. These formulas are used to calculate the probability of compound events, distinguish between mutually exclusive and independent events, and determine the number of possible outcomes in various scenarios.
- 4.8: Chapter 4 - Key Terms and Symbols
- This section provides key symbols and definitions that form the foundation for understanding Chapter 4: Probability and Counting Rules. It introduces essential terms such as experiment, outcome, event, and sample space, along with counting rule definitions. These definitions help readers interpret problems correctly and apply the appropriate counting techniques, such as the multiplication rule, permutations, and combinations. By familiarizing readers with these core concepts and symbols, the sec