Probability and Counting Principles: Cheat Sheet

Essential Concepts

If one event can occur in [latex]m[/latex] ways and a second event with no common outcomes can occur in [latex]n[/latex] ways, then the first or second event can occur in [latex]m+n[/latex] ways.
If one event can occur in [latex]m[/latex] ways and a second event can occur in [latex]n[/latex] ways after the first event has occurred, then the two events can occur in [latex]m\times n[/latex] ways.
A permutation is an ordering of [latex]n[/latex] objects.
If we have a set of [latex]n[/latex] objects and we want to choose [latex]r[/latex] objects from the set in order, we write [latex]P\left(n,r\right)[/latex].
Permutation problems can be solved using the Multiplication Principle or the formula for [latex]P\left(n,r\right)[/latex].
A selection of objects where the order does not matter is a combination.
Given [latex]n[/latex] distinct objects, the number of ways to select [latex]r[/latex] objects from the set is [latex]\text{C}\left(n,r\right)[/latex] and can be found using the formula: [latex]C\left(n,r\right)=\dfrac{n!}{r!\left(n-r\right)!}[/latex].
A set containing [latex]n[/latex] distinct objects has [latex]{2}^{n}[/latex] subsets.
For counting problems involving non-distinct objects, we need to divide to avoid counting duplicate permutations.

[latex]\left(\begin{gathered}n\\ r\end{gathered}\right)[/latex] is called a binomial coefficient and is equal to [latex]C\left(n,r\right)[/latex].
The Binomial Theorem allows us to expand binomials [latex](x+y)^n[/latex] without multiplying. That is: [latex]{\left(x+y\right)}^{n}=\sum\limits _{k - 0}^{n}\left(\begin{gathered}n\\ k\end{gathered}\right){x}^{n-k}{y}^{k}[/latex]
We can find a given term of a binomial expansion without fully expanding the binomial.
The [latex]\left(r+1\right)th[/latex] term of a binomial expansion is [latex]\left(\begin{gathered}n\\ r\end{gathered}\right){x}^{n-r}{y}^{r}[/latex].

Probability is always a number between [latex]0[/latex] and [latex]1[/latex], where [latex]0[/latex] means an event is impossible and 1 means an event is certain.
The probabilities in a probability model must sum to [latex]1[/latex].
When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in the sample space for the experiment. That is: [latex]P\left(E\right)=\dfrac{n\left(E\right)}{n\left(S\right)}[/latex]
To find the probability of the union of two events, we add the probabilities of the two events and subtract the probability that both events occur simultaneously. The formula is [latex]P\left(E\cup F\right)=P\left(E\right)+P\left(F\right)-P\left(E\cap F\right)[/latex].
To find the probability of the union of two mutually exclusive events, we add the probabilities of each of the events. The formula is [latex]P\left(E\cup F\right)=P\left(E\right)+P\left(F\right)[/latex].
The probability of the complement of an event is the difference between [latex]1[/latex] and the probability that the event occurs.
In some probability problems, we need to use permutations and combinations to find the number of elements in events and sample spaces.

number of permutations of [latex]n[/latex] distinct objects taken [latex]r[/latex] at a time	[latex]P\left(n,r\right)=\dfrac{n!}{\left(n-r\right)!}[/latex]
number of combinations of [latex]n[/latex] distinct objects taken [latex]r[/latex] at a time	[latex]C\left(n,r\right)=\dfrac{n!}{r!\left(n-r\right)!}[/latex]
number of permutations of [latex]n[/latex] non-distinct objects	[latex]\dfrac{n!}{{r}_{1}!{r}_{2}!\dots {r}_{k}!}[/latex]
Binomial Theorem	[latex]{\left(x+y\right)}^{n}=\sum\limits _{k - 0}^{n}\left(\begin{gathered}n\\ k\end{gathered}\right){x}^{n-k}{y}^{k}[/latex]
[latex]\left(r+1\right)th[/latex] term of a binomial expansion	[latex]\left(\begin{gathered}n\\ r\end{gathered}\right){x}^{n-r}{y}^{r}[/latex]
probability of an event with equally likely outcomes	[latex]P\left(E\right)=\dfrac{n\left(E\right)}{n\left(S\right)}[/latex]
probability of the union of two events	[latex]P\left(E\cup F\right)=P\left(E\right)+P\left(F\right)-P\left(E\cap F\right)[/latex]
probability of the union of mutually exclusive events	[latex]P\left(E\cup F\right)=P\left(E\right)+P\left(F\right)[/latex]
probability of the complement of an event	[latex]P\left(E\text{'}\right)=1-P\left(E\right)[/latex]

addition principle: if one event can occur in [latex]m[/latex] ways and a second event with no common outcomes can occur in [latex]n[/latex] ways, then the first or second event can occur in [latex]m+n[/latex] ways

binomial coefficient: the number of ways to choose [latex]r[/latex] objects from [latex]n[/latex] objects where order does not matter; equivalent to [latex]C\left(n,r\right)[/latex], denoted [latex]\left(\begin{gathered}n\\ r\end{gathered}\right)[/latex]

binomial expansion: the result of expanding [latex]{\left(x+y\right)}^{n}[/latex] by multiplying

complement of an event: the set of outcomes in the sample space that are not in the event [latex]E[/latex]

fundamental counting principle: if one event can occur in [latex]m[/latex] ways and a second event can occur in [latex]n[/latex] ways after the first event has occurred, then the two events can occur in [latex]m\times n[/latex] ways; also known as the Multiplication Principle

multiplication principle: if one event can occur in [latex]m[/latex] ways and a second event can occur in [latex]n[/latex] ways after the first event has occurred, then the two events can occur in [latex]m\times n[/latex] ways; also known as the Fundamental Counting Principle

permutation
a selection of objects in which order matters

probability: a number from [latex]0[/latex] to [latex]1[/latex] indicating the likelihood of an event

probability model: a mathematical description of an experiment listing all possible outcomes and their associated probabilities