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Introduction to Probability
Uncertainty, Probability, Tree Diagrams,
Combinations and Permutations
Chapter 4
BA 201
Slide 1
PROBABILITY
Slide 2
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?
Slide 3
Probability
Probability is a numerical measure of the likelihood
that an event will occur.
Probability values are from 0 to 1.
Slide 4
Probability as a Numerical Measure
of the Likelihood of Occurrence
Increasing Likelihood of Occurrence
Probability:
0
The event
is very
unlikely
to occur.
0.5
The occurrence
of the event is
just as likely as
it is unlikely.
1
The event
is almost
certain
to occur.
Slide 5
STATISTICAL EXPERIMENTS
Slide 6
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.
Slide 7
An Experiment and Its Sample Space
An experiment is any process that generates welldefined 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
Slide 8
An Experiment and Its Sample Space
Experiment
Experiment Outcomes
Toss a coin
Head, tail
Inspect a part
Defective, non-defective
Conduct a sales call
Purchase, no purchase
Slide 9
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)
Collins Mining
Markley Oil
8
10
-2
5
0
-20
Slide 10
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)
Slide 11
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:
Collins Mining:
n1 = 4
n2 = 2
Total Number of
Experimental Outcomes:
n1n2 = (4)(2) = 8
Slide 12
Tree Diagram
Bradley Investments
Markley Oil
(Stage 1)
Collins Mining
(Stage 2)
Gain 8
Gain 10
Gain 8
Gain 5
Lose 2
Lose 2
Gain 8
Even
Lose 20
Gain 8
Lose 2
Lose 2
Experimental
Outcomes
(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
Slide 13
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.
CnN
where:
N!
 N
  
 n  n !(N - n )!
N! = N(N - 1)(N - 2) . . . (2)(1)
n! = n(n - 1)(n - 2) . . . (2)(1)
0! = 1
Slide 14
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.
PnN
where:
N!
 N
 n !  
 n  (N - n )!
N! = N(N - 1)(N - 2) . . . (2)(1)
n! = n(n - 1)(n - 2) . . . (2)(1)
0! = 1
Slide 15
Combinations and Permutations
4 Objects: A B C D
C24 
N!
4!

n!( N - n)! 2!(4 - 2)!
4!
24
24



6
2!*2! 2 * 2 4
AB
BC
AC
BD
AD
CD
P24 
N!
4!
4! 24

 
 12
( N - n)! (4 - 2)! 2! 2
AB
BA
AC
CA
AD
DA
BC
CB
BD
DB
CD
DC
Slide 16
PRACTICE
TREE DIAGRAMS,
COMBINATIONS, AND
PERMUTATIONS
Slide 17
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.
Slide 18
Combinations
There are five boxes numbered 1 through 5. You pick two
boxes.
How many combinations of boxes are there?
Show the combinations.
C
N
n
N!

n!( N - n)!
Slide 19
Combinations
There are five boxes numbered 1 through 5. You pick two
boxes.
How many permutations of boxes are there?
Show the permutations.
N!
P 
( N - n)!
N
n
Slide 20
Slide 21