# Probability and Independence

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

For an experiment we define an event to be any collection of possible outcomes. A simple event is an event that consists of exactly one outcome.

• "or" means the union (i.e. either can occur)
• "and" means intersection (i.e. both must occur)

Two events are mutually exclusive if they cannot occur simultaneously. For a Venn diagram, we can tell that two events are mutually exclusive if their regions do not intersect

##### Definition: Probability

We define probability of an event $$E$$ to be to be

$P(E) = \dfrac{\text{number of simple events within E}}{\text{ total number of possible outcomes}} \nonumber$

We have the following:

1. $$P(E)$$ is always between 0 and 1.
2. The sum of the probabilities of all simple events must be 1.
3. $$P(E) + P(\text{not } E) = 1$$
4. If $$E$$ and $$F$$ are mutually exclusive then

$P(E \text{ or } F) = P(E) + P(F) \nonumber$

## The Difference Between "and" and "or"

If $$E$$ and $$F$$ are events then we use the terminology

$E \text{ and } F \nonumber$

to mean all outcomes that belong to both $$E$$ and $$F$$.

We use the terminology

$E \text{ or } F \nonumber$

to mean all outcomes that belong to either $$E$$ or $$F$$.

Below is an example of two sets, $$A$$ and $$B$$, graphed in a Venn diagram.

The green area represents $$A$$ and $$B$$ while all areas with color represent $$A$$ or $$B$$

##### Example $$\PageIndex{1}$$

Our Women's Volleyball team is recruiting for new members. Suppose that a person inquires about the team.

• Let $$E$$ be the event that the person is female
• Let $$F$$ be the event that the person is a student

then $$E$$ and $$F$$ represents the qualifications for being a member of the team. Note that $$E$$ or $$F$$ is not enough.

We define

##### Definition: Conditional Probability

$P(E|F) = \dfrac{P(E \text{ and } F}{P(F)} \nonumber$

We read the left hand side as "The probability of event $$E$$ given event $$F$$ occurred."

We call two events independent if the following definitions hold.

##### Definition: Independence

For independent Events

$P(E|F) = P(E) \label{1a}$

Equivalently, we can say that $$E$$ and $$F$$ are independent if

##### Definition: The Multiplication Rule

For Independent Events

$P(E \text{ and } F) = P(E)P(F) \label{1b}$

##### Example $$\PageIndex{2}$$

Consider rolling two dice. Let

• $$E$$ be the event that the first die is a 3.
• $$F$$ be the event that the sum of the dice is an 8.

Then $$E$$ and $$F$$ means that we rolled a three and then we rolled a 5

This probability is 1/36 since there are 36 possible pairs and only one of them is (3,5)

We have

$P(E) = 1/6 \nonumber$

And note that (2,6),(3,5),(4,4),(5,3), and (6,2) give $$F$$

Hence

$P(F) = 5/36 \nonumber$

We have

$P(E) P(F) = (1/6) (5/36) \nonumber$

which is not 1/36 and we can conclude that $$E$$ and $$F$$ are not independent.

##### Exercise $$\PageIndex{2}$$

Test the following two events for independence:

• $$E$$ the event that the first die is a 1.
• $$F$$ the event that the sum is a 7.

## A Counting Rule

For two events, $$E$$ and $$F$$, we always have

$P(E \text{ or } F) = P(E) + P(F) - P(E \text{ and } F) \label{2}$

##### Example $$\PageIndex{3}$$

Find the probability of selecting either a heart or a face card from a 52 card deck.

###### Solution

We let

• $$E$$ = the event that a heart is selected
• $$F$$ = the event that a face card is selected

then

$P(E) = \dfrac{1}{4} \nonumber$

and

$P(F) = \dfrac{3}{13} \nonumber$

that is, Jack, Queen, or King out of 13 different cards of one kind.

$P(E \text{ and } F) = \dfrac{3}{52} \nonumber$

The counting rule formula (eq. 2) gives

$P(E \text{ or } F) = \dfrac{1}{4} + \dfrac{3}{13} - \dfrac{3}{52} = \dfrac{22}{52} = 42\text{%} \nonumber$

### Contributors

This page titled Probability and Independence is shared under a not declared license and was authored, remixed, and/or curated by Larry Green.