# More Probability Concepts

Need an introduction to probability first? Basic Probability Concepts

## Properties of Probabilities

Recall that the probability of an event $\,E\,$ is denoted by $\,P(E)\,.$

### Probabilities are Always Numbers Between $\,0\,$ and $\,1\,$

For any event $\,E\,,$ $\,0\le P(E)\le 1\,.$ That is, a probability is always a number between $\,0\,$ and $\,1\,.$

### The Sample Space has Probability $\,1\,$

If $\,S\,$ is the sample space,
then $\,P(S)=1\,.$
That is, there's a $\,100\%\,$
chance that *something* will happen.

### The Meaning of Probabilities $\,0\,$ and $\,1\,$ for Finite Sample Spaces

For a finite sample space
(i.e., an experiment with a finite number of outcomes):
a probability of $\,1\,$ means
the event will definitely occur;
a probability of $\,0\,$ means the
event will definitely *not* occur.

Notice that if the experiment
is to randomly choose a real number
from the interval $\,[0,1]\,,$
then the probability of
choosing (say) the number $\,0.5\,$ is zero,
and yet this outcome could occur.
This is why a *finite*
sample space is needed in the above statement.

### Probability of a Union

For any events $\,A\,$ and $\,B\,$: $$\cssId{s17}{P(A\cup B)} \cssId{s18}{=P(A)+P(B)-P(A\cap B)}$$ If you were just to add the two probabilities, then the intersection gets added twice; this extra probability must be subtracted.

### Probability of the Complement

Recall that $\,\overline{E}\,$ denotes the complement of $\,E\,.$

For any event, either it happens, or it doesn't!

Therefore: $$P(E)+P(\overline{E})=1$$

Equivalently: $$P(\overline{E})=1 -P(E)$$

### Mutually Exclusive Events

Recall that $\,\emptyset \,$ denotes the empty set.

Events $\,A\,$ and $\,B\,$ are said to be
*mutually exclusive* if
$\,A\cap B = \emptyset\,.$
That is, two events
are mutually exclusive when they
have nothing in common
(their intersection is empty).

Since $\,P(\emptyset) = 0\,,$ for mutually exclusive events, the above formula is simpler: $$\cssId{s33}{\,P(A\cup B)=P(A)+P(B)}\,$$

## Multiplication Counting Principle

If there are $\,F\,$ choices for how to perform a first act, and for each of these $\,F\,$ ways, there are $\,S\,$ choices for how to perform a second act, then there are $\,F\cdot S\,$ ways to perform the acts in succession. (The idea extends to more than $\,2\,$ acts.)

The idea is illustrated by the diagram above. If there are $\,2\,$ piles, with $\,3\,$ in each pile, then the total is $\,2\cdot 3 = 6\,.$

## Example (Pizza Choices)

Suppose that a pizza shop offers $\,3\,$ types of crust, $\,2\,$ different types of cheese, and $\,7\,$ different toppings.

A single-topping pizza consists of a choice of crust, a choice of cheese (if desired), and a choice of topping.

How many different single-topping pizzas are there?

Solution: There are $\,3\,$ choices for the crust, $\,3\,$ choices for cheese (none, first type, second type), and $\,7\,$ choices for the single topping.

By the Multiplication Counting Principle, there are $\,3\cdot 3\cdot 7 = 63\,$ possible single-topping pizzas.