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 1-3, 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.