1
Basic Axioms of Probability:
Lecture II
I.
Basics of Probability
A.
Using the example from Birenens Chapter 1: Assume we are interested in the
game Texas lotto (similar to Florida lotto).
1.
In this game, players choose a set of 6 numbers out of the first 50.
Note that the ordering does not count so that 35,20,15,1,5,45 is the
same of 35,5,15,20,1,45.
2.
How many different sets of numbers can be drawn?
a.
First, we note that we could draw any one of 50 numbers in the
first draw.
b.
However for the second draw we can only draw 49 possible
numbers (one of the numbers has been eliminated). Thus, there are
50 x 49 different ways to draw two numbers
c.
Again, for the third draw, we only have 48 possible numbers left.
Therefore, the total number of possible ways to choose 6 numbers
out of 50 is
50
5
50
1
50 6
1
45
1
50!
50
50 6 !
k
jk
k
k
k
d.
Finally, note that there are 6! ways to draw a set of 6 numbers (you
could draw 35 first, or 20 first, …). Thus, the total number of ways
to draw an unordered set of 6 numbers out of 50 is
50
50!
15,890,700
6
6! 50 6 !
e.
This is a combinatorial. It also is useful for binomial arithmetic:
0
n
n
k
n k
k
n
a
b
a b
k
B.
Definitions:
1.
Sample Space
: The set of all possible outcomes. In the Texas lotto
scenario, the sample space is all possible 15,890,700 sets of 6 numbers
which could be drawn.
2.
Event
: A subset of the sample space. In the Texas lotto scenario,
possible events include single draws such as {35,20,15,1,5,45} or
complex draws such as all possible lotto tickets including {35,20,15}.
Note that this could be {35,20,15,1,2,3}, {35,20,15,1,2,4},….
3.
Simple Event
: An event which cannot be a union of other events. In the
Texas lotto scenario, this is a single draw such as {35,20,15,1,5,45}.
4.
Composite Event
: An event which is not a simple event.