1. Introduction to Probability Uncertainty, Probability, Tree Diagrams, Combinations and Permutations
Chapter 4
BA 201

2. Probability

3. Uncertainty Managers often base their decisions on an analysis
of uncertainties such as the following:
What are the chances that sales will decrease
if we increase prices?
What is the likelihood a new assembly method
method will increase productivity?
What are the odds that a new investment will
be profitable?

4. Probability Probability is a numerical measure of the likelihood
that an event will occur.
Probability values are from 0 to 1.

5. Probability as a Numerical Measure of the Likelihood of Occurrence
Increasing Likelihood of Occurrence
0.5 1
Probability:
The event
is very
unlikely
to occur.
The occurrence
of the event is
just as likely as
it is unlikely.
The event
is almost
certain
to occur.

6. Statistical Experiments

7. Statistical Experiments
In statistical experiments, probability determines
outcomes.
Even though the experiment is repeated in exactly
the same way, an entirely different outcome may
occur.

8. An Experiment and Its Sample Space
An experiment is any process that generates well-
defined outcomes.
The sample space for an experiment is the set of
all experimental outcomes.
An experimental outcome is also called a sample point.
Roll a die 1 2 3 4 5 6

9. An Experiment and Its Sample Space
Toss a coin
Inspect a part
Conduct a sales call
Experiment Outcomes
Head, tail
Defective, non-defective
Purchase, no purchase

10. An Experiment and Its Sample Space
Bradley Investments
Bradley has invested in two stocks, Markley Oil
and Collins Mining. Bradley has determined that the
possible outcomes of these investments three months
from now are as follows.
Investment Gain or Loss
in 3 Months (in $000)
Markley Oil
Collins Mining 10 5
-20 8 -2

11. A Counting Rule for Multiple-Step Experiments
If an experiment consists of a sequence of k steps in which there are n1 possible results for the first step, n2 possible results for the second step, and so on, then the total number of experimental outcomes is given by:
# outcomes = (n1)(n2) (nk)

12. A Counting Rule for Multiple-Step Experiments
Bradley Investments
Bradley Investments can be viewed as a two-step
experiment. It involves two stocks, each with a set of
experimental outcomes.
Markley Oil: n1 = 4
Collins Mining: n2 = 2
Total Number of
Experimental Outcomes: n1n2 = (4)(2) = 8

13. Tree Diagram Bradley Investments Markley Oil Collins Mining
(Stage 1)
Collins Mining
(Stage 2)
Experimental
Outcomes
Gain 8
Lose 2
(10, 8) Gain $18,000
(10, -2) Gain $8,000
(5, 8) Gain $13,000
(5, -2) Gain $3,000
(0, 8) Gain $8,000
(0, -2) Lose $2,000
(-20, 8) Lose $12,000
(-20, -2) Lose $22,000
Gain 5
Gain 10
Lose 20
Even

14. Counting Rule for Combinations
Number of Combinations of N Objects Taken n at a Time
Combinations enable us to count the number of experimental outcomes when n objects are to be selected from a set of N objects.
Order does not matter – which means a combination lock is really a misnomer.
Without replacement/repetition.
where: N! = N(N - 1)(N - 2) (2)(1)
n! = n(n - 1)(n - 2) (2)(1)
0! = 1

15. Counting Rule for Permutations
Number of Permutations of N Objects Taken n at a Time
Permutations enable us to count the number of experimental outcomes when n objects are to be selected from a set of N objects, where the order of selection is important.
Order matters – so why is it not called a permutation lock?
Without replacement/repetition.
where: N! = N(N - 1)(N - 2) (2)(1)
n! = n(n - 1)(n - 2) (2)(1)
0! = 1

16. Combinations and Permutations
4 Objects: A B C D
A B
B A
A C
C A
A B
B C
A D
D A
A C
B D
B C
C B
B D
D B
A D
C D
C D
D C

17. Practice Tree Diagrams, Combinations, and Permutations

18. Practice Tree Diagram
A box contains six balls: two green, two blue, and two red.
You draw two balls without looking.
How many outcomes are possible?
Draw a tree diagram depicting the possible outcomes.
GG GB GR
RG RB RR
BG BB BR

19. Combinations
There are five boxes numbered 1 through 5. You pick two boxes.
How many combinations of boxes are there?
Show the combinations.
4 5

20. Combinations
There are five boxes numbered 1 through 5. You pick two boxes.
How many permutations of boxes are there?
Show the permutations.