4.7: Chapter 4 Formulas
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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 Formulas
General Formula for the Probability of an Event
The probability that an event \(E\) will occur is the ratio of the number of outcomes in the event to the total number of outcomes in the sample space. It gives a numerical measure of the likelihood that the event will happen.
\( P(E) = \dfrac{\text{number of outcomes in the event}}{\text{total number of outcomes}} \)
Addition Rule #I for Probability for Mutually Exclusive Events
The Addition Rule I in probability theory is used to find the probability that one of two events occurs when the events are mutually exclusive. Mutually exclusive events cannot occur at the same time. This means that if events A and B cannot happen at the same time, the probability of either A or B happening is the sum of their probabilities.
\( P(A \text{ or } B) = P(A) + P(B) \)
Addition Rule #II for Probability for Non-Mutually Exclusive Events
The Addition Rule II in probability theory is used to find the probability that one of two events occurs, even if the events are not mutually exclusive. When events can occur at the same time, we must subtract the probability of both events occurring together to avoid double-counting.
\( P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \)
At Least One Probability
The "at least one" rule in probability is used to calculate the probability that an event occurs at least once in multiple trials. Rather than computing all possible ways the event can happen, it is often simpler to use the complement rule: subtract the probability that the event does not occur at all from 1.
\( P(\text{at least one}) = 1 - P(\text{none}) \)
Complement Rule for Probability
The complement rule in probability is used to find the probability that an event does not occur. It is based on the fact that the total probability of all possible outcomes is 1. Therefore, the probability that event A does not happen is equal to 1 minus the probability that it does happen.
\( P(\text{Not } A) = 1 - P(A) \)
Conditional Probability
Conditional probability is the probability that one event occurs given that another event has already occurred. It helps us understand how the occurrence of one event affects the likelihood of another. This means the probability of B given A is equal to the probability of both A and B occurring divided by the probability of A.
\( P(B \mid A) = \dfrac{P(A \text{ and } B)}{P(A)} \)
Multiplication Rule I for Probability for Independent Events
The Multiplication Rule I in probability theory is used to find the probability that two events occur in sequence, assuming the events are independent. Independent events are those where the outcome of one event does not affect the outcome of the other.
\( P(A \text{ and } B) = P(A) \cdot P(B) \)
Multiplication Rule II for Probability for Dependent Events
The Multiplication Rule II in probability theory is used to find the probability that two events occur in sequence when the events are dependent. Dependent events are those where the outcome of the first event affects the outcome of the second event. This means the probability of both A and B occurring is equal to the probability of A times the probability of B, given that A has occurred.
\( P(A \text{ and } B) = P(A) \cdot P(B \mid A) \)
Combination Rule
The combination rule is used to determine the number of ways to choose \(r\) items from a group of \(n\) items when the order of selection does not matter. This is useful in situations where the arrangement of selected items is irrelevant.
\( {}_nC_r = \dfrac{n!}{r!(n - r)!} \)
Fundamental Counting Rule
The fundamental counting rule is used to determine the total number of possible outcomes when a sequence of events occurs and each event has a specific number of possible outcomes. If the first event can occur in \(m_1\) ways, the second in \(m_2\) ways, and so on, then the total number of combined outcomes is the product of all these values.
\( \text{Total outcomes} = m_1 \cdot m_2 \cdot m_3 \cdots m_k \)
n-Factorial
The term "n factorial" (written as \(n!\)) represents the product of all positive integers from 1 to \(n\). It is used frequently in counting and probability problems, especially in permutations and combinations.
\( n! = n \cdot (n - 1) \cdot (n - 2) \cdot \dots \cdot 2 \cdot 1 \)
Permutation of \(n\) Distinct Objects
The permutation of \(n\) items is an arrangement of those items in a specific order. If all \(n\) items are used, the total number of different permutations is computed using the formula below.
\(n!\)
Permutation with Repetition of Identical Items
The permutation rule with objects that repeat is used when some objects are identical (i.e., indistinguishable from each other); the total number of distinct permutations is reduced. The formula accounts for repeated items by dividing by the factorial of each group of repeated objects, where there are \(n\) total items, with \(r_1\) indistinguishable of one kind, \(r_2\) of another, and so on is provided below.
\( \dfrac{n!}{r_1! \cdot r_2! \cdot \dots \cdot r_k!} \)
Permutation Rule for Selecting \(r\) Objects from \(n\) Distinct Items
The permutation rule for \(n\) choose \(r\) gives the number of ways to arrange \(r\) items from a set of \(n\) distinct items, where the order matters and no repetitions are allowed. The number of such permutations is computed using the formula below.
\( {}_nP_r = \dfrac{n!}{(n - r)!} \)
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"4.7: Chapter 4 Formulas" by Toros Berberyan, Tracy Nguyen, and Alfie Swan is licensed under CC BY 4.0