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Contemporary Mathematics

Formula Review

Contemporary MathematicsFormula Review

Formula Review

7.2 Permutations

${}_{n}P_{r} = \frac{n!}{(n - r)!}$

7.3 Combinations

The formula for counting combinations is:
$_{n} C_{r} = \frac{n!}{r! (n - r)!}$

7.5 Basic Concepts of Probability

For an experiment whose sample space $S$ consists of equally likely outcomes, the theoretical probability of the event $E$ is the ratio $P (E) = \frac{n (E)}{n (S)}$ where $n (E)$ and $n (S)$ denote the number of outcomes in the event and in the sample space, respectively.
$P (E) = 1 - P (E')$

7.7 What Are the Odds?

For an event $E$ ,
$\begin{matrix} odds for E = n (E) : n (E^{'}) = P (E) : P (E^{'}) = P (E) : (1 - P (E)) \\ odds against E = n (E^{'}) : n (E) = P (E^{'}) : P (E) = (1 - P (E)) : P (E) \end{matrix}$
If the odds in favor of $E$ are $A : B$ , then $P (E) = \frac{A}{A + B}$ .

7.8 The Addition Rule for Probability

If $E$ and $F$ are mutually exclusive events, then
$P (E or F) = P (E) + P (F)$ .
If $E$ and $F$ are events that contain outcomes of a single experiment, then
$P (E or F) = P (E) + P (F) - P (E and F)$ .

7.9 Conditional Probability and the Multiplication Rule

If $E$ and $F$ are events associated with the first and second stages of an experiment, then $P (E and F) = P (E) \times P (F | E)$ .

7.10 The Binomial Distribution

Suppose we have a binomial experiment with $n$ trials and the probability of success in each trial is $p$ . Then:
$P (number of successes is a) = {}_{n}C_{a} \times p^{a} \times {(1 - p)}^{n - a}$

7.11 Expected Value

If $O$ represents an outcome of an experiment and $n (O)$ represents the value of that outcome, then the expected value of the experiment is:
$\sum n (O) \times P (O)$

where $Σ$ stands for the sum, meaning we add up the results of the formula that follows over all possible outcomes.

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- Authors: Donna Kirk
- Publisher/website: OpenStax
- Book title: Contemporary Mathematics
- Publication date: Mar 22, 2023
- Location: Houston, Texas
- Book URL: https://openstax.org/books/contemporary-mathematics/pages/1-introduction
- Section URL: https://openstax.org/books/contemporary-mathematics/pages/7-formula-review

© Dec 21, 2023 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.