Contemporary Mathematics

# Formula Review

## 7.2Permutations

• $Pnr=n!(n−r)!Pnr=n!(n−r)!$

## 7.3Combinations

• The formula for counting combinations is:
$nCr=n!r!(n−r)!nCr=n!r!(n−r)!$

## 7.5Basic Concepts of Probability

• For an experiment whose sample space $SS$ consists of equally likely outcomes, the theoretical probability of the event $EE$ is the ratio $P(E)=n(E)n(S)P(E)=n(E)n(S)$ where $n(E)n(E)$ and $n(S)n(S)$ denote the number of outcomes in the event and in the sample space, respectively.
• $P(E)=1−P(E′)P(E)=1−P(E′)$

## 7.7What Are the Odds?

• For an event $EE$,
$odds forE=n(E):n(E′)=P(E):P(E′)=P(E):(1−P(E))odds againstE=n(E′):n(E)=P(E′):P(E)=(1−P(E)):P(E)odds forE=n(E):n(E′)=P(E):P(E′)=P(E):(1−P(E))odds againstE=n(E′):n(E)=P(E′):P(E)=(1−P(E)):P(E)$
• If the odds in favor of $EE$ are $A:BA:B$, then $P(E)=AA+BP(E)=AA+B$.

## 7.8The Addition Rule for Probability

• If $EE$ and $FF$ are mutually exclusive events, then
$P(EorF)=P(E)+P(F)P(EorF)=P(E)+P(F)$.
• If $EE$ and $FF$ are events that contain outcomes of a single experiment, then
$P(EorF)=P(E)+P(F)−P(E⁢and⁢F)P(EorF)=P(E)+P(F)−P(E⁢and⁢F)$.

## 7.9Conditional Probability and the Multiplication Rule

• If $EE$ and $FF$ are events associated with the first and second stages of an experiment, then $P(Eand⁢F)=P(E)×P(F|E)P(Eand⁢F)=P(E)×P(F|E)$.

## 7.10The Binomial Distribution

• Suppose we have a binomial experiment with $nn$ trials and the probability of success in each trial is $pp$. Then:
$P(number of successes isa)=Cna×pa×(1−p)n−aP(number of successes isa)=Cna×pa×(1−p)n−a$

## 7.11Expected Value

• If $OO$ represents an outcome of an experiment and $n(O)n(O)$ represents the value of that outcome, then the expected value of the experiment is:
$∑ n(O)×P(O)∑ n(O)×P(O)$

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

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