Probability and Independence

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

• Larry Green (Lake Tahoe Community College)