1
Recap…
Axioms and Properties of Probability
Axiom:
A1) P(
S
) = 1
A2) P(
E
)
≥
0
for every event
E
S
A3) For mutually exclusive events,
Properties:
1) P(
) = 0
2)
3)
4)
Today: Simple Rules for counting the number
of “simple” outcomes in an event
1.
Counting the Number of Operations - Product Rule (Page 60-61)
If an operation consists of k steps, and there are
n
1
choices for the
first step, ...,
n
k
choices for the
k
th
step, then the total number of
operations possible is the product
n
1
×
n
2
×
×
n
k
Logic: Use Tree Diagram
2.
Basic Problem: Select a subgroup of size k from n objects.
•
How many possible ways?
Answer depends on “rules”:
1.
Order: Matters (permutation) or
doesn’t matter (Combination)
2.
Replacement: Yes or No.
•
So four types of situations
2
Section 2.3

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2
Counting Techniques Useful for
Probability Calculations
Focus: Situation where
S
is finite and has
N
outcomes, and all
N
simple
events are equally likely.
Let
A
be an event, and
N(A)
be the number of outcomes in
A
:
3
Section 2.3
For a small sample space,
you can enumerate all outcomes and
then count manually.
Example: Roll two dice and sum them:
N
(
S
) = 36
simple events
Not always easy to enumerate manually.
4
Section 2.3