## Introduction to Probability

### Sample Spaces

A sample space is the set of all possible outcomes. However, some sample spaces are better than others.

Consider the experiment of flipping two coins. It is possible to get 0 heads, 1 head, or 2 heads. Thus, the sample space could be {0, 1, 2}. Another way to look at it is flip { HH, HT, TH, TT }. The second way is better because each event is as equally likely to occur as any other.

When writing the sample space, it is highly desirable to have events which are equally likely.

Another example is rolling two dice. The sums are { 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 }. However, each of these aren't equally likely. The only way to get a sum 2 is to roll a 1 on both dice, but you can get a sum of 4 by rolling a 3-1, 2-2, or 3-1. The following table illustrates a better sample space for the sum obtain when rolling two dice.
 First Die Second Die 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12

### Classical Probability

The above table lends itself to describing data another way -- using a probability distribution. Let's consider the frequency distribution for the above sums.
 Sum Frequency Relative Frequency 2 1 1/36 3 2 2/36 4 3 3/36 5 4 4/36 6 5 5/36 7 6 6/36 8 5 5/36 9 4 4/36 10 3 3/36 11 2 2/36 12 1 1/36

If just the first and last columns were written, we would have a probability distribution. The relative frequency of a frequency distribution is the probability of the event occurring. This is only true, however, if the events are equally likely.

This gives us the formula for classical probability. The probability of an event occurring is the number in the event divided by the number in the sample space. Again, this is only true when the events are equally likely. A classical probability is the relative frequency of each event in the sample space when each event is equally likely.

P(E) = n(E) / n(S)

### Empirical Probability

Empirical probability is based on observation. The empirical probability of an event is the relative frequency of a frequency distribution based upon observation.

P(E) = f / n

### Probability Rules

There are two rules which are very important.

#### All probabilities are between 0 and 1 inclusive

0 <= P(E) <= 1

#### The sum of all the probabilities in the sample space is 1

There are some other rules which are also important.

#### The probability of an event which cannot occur is 0.

The probability of any event which is not in the sample space is zero.

#### The probability of an event which must occur is 1.

The probability of the sample space is 1.

#### The probability of an event not occurring is one minus the probability of it occurring.

P(E') = 1 - P(E)