3.8: Chapter Review
- Page ID
- 36660
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)3.1 Introduction
Key Terms:
experiment: an activity or process that has a set of well-defined results and can be repeated indefinitely
outcomes: the results of an experiment
sample space: the collection of all possible outcomes of the experiment
event: a set of certain outcomes of an experiment that you want to have happen
tree diagram: a graphical way of representing a random experiment with multiple steps using branches for outcomes
3.2 Three Types of Probability
Key Terms:
probability: the likelihood of an event happening
theoretical probability (classical approach): the probability calculated from the number of favorable outcomes divided by the total number of outcomes when each outcome has an equal probability
empirical (experimental) probability: the probability calculated by finding the relative frequency of an event from performing an experiment many times
Law of Large Numbers: as n increases, the relative frequency tends toward the theoretical probability
subjective probability: the probability of an event is estimated using previous knowledge and is someone’s opinion
Formulas:
Theoretical Probability: \(P(A) = \dfrac{\text{Number of ways A can occur}}{\text{Number of different outcomes in S}}\)
Empirical Probability: \(P(A) = \dfrac{\text{Number of times A occurred}}{\text{Number of times the experiment was repeated}}\)
3.3 Complement Rule
Key Terms:
complementary events: events that have no outcomes in common and together make up the entire sample space
Venn diagram: a visual way to represent sets and probability using a rectangle to represent the sample space and circles to represent events
Formulas:
Complement Rule: P(A) + P(A’) = 1 or P(A) = 1 – P(A’) or P(A’) = 1 – P(A)
3.4 Union and Intersection
Key Terms:
mutually exclusive (disjoint) events: events that cannot occur at the same time
intersection: where two events overlap and happen at the same time
union: the junction of two events including their intersection
Formulas:
Mutually Exclusive Events: \(P(A \cap B) = 0\)
Addition Rule for Not Mutually Exclusive Events: \(P(A \cup B) = P(A) + P(B) - P(A \cap B) \)
Addition Rule for Mutually Exclusive Events: \(P(A \cup B) = P(A) + P(B) \)
3.5 Independent Events
Key Terms:
independent events: two events that are not related and the outcome of one event does not affect the probability of the other event
dependent events: two events that are related and the outcome of one event does affect the probability of the other event
Formulas:
Multiplication Rule for Independent Events: \(P(A \cap B) = P(A) \cdot P(B) \)
Probability of “at least one”: P(at least one) = 1 – P(none)
3.6 Conditional Probability
Key Terms:
conditional probability: the probability of an event happening, given that another event already happened
Formulas:
General Multiplication Rule: \(P(A \cap B) = P(A) \cdot P(B|A) \)
Conditional Probability Rule: \(P(A|B) = \dfrac{P(A \cap B)}{P(B)} \) or \(P(B|A) = \dfrac{P(A \cap B)}{P(A)} \)
3.7 Counting Rules
Key Terms:
Fundamental Counting Rule: the number of ways to do event 1, 2, … n together would be to multiply the number of ways each event can be done, m1 · m2 · … · mn
factorial: the mathematical way to multiply a list of decreasing numbers
permutation: an arrangement of items with a specific order
combination: an arrangement of items when order is not important
Formulas:
Factorial Rule: \(n! = n(n – 1)(n – 2) \ldots \cdot 3 \cdot 2 \cdot 1\)
Permutation Rule: \( {}_n P_{r} = P(n, r) = \dfrac{n!}{(n - r)!}\)
Combination Rule: \( {}_n C_{r} = C(n, r) = \dfrac{n!}{r!(n - r)!}\)