Formula Review - Contemporary Mathematics

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.