1 Slide
Introduction to ProbabilityUncertainty, Probability, Tree Diagrams,
Combinations and Permutations
Chapter 4BA 201
2 Slide
PROBABILITY
3 Slide
Uncertainty
Managers often base their decisions on an analysis of uncertainties such as the following:
What are the chances that sales will decreaseif we increase prices?What is the likelihood a new assembly method method will increase productivity?What are the odds that a new investment willbe profitable?
4 Slide
Probability
Probability is a numerical measure of the likelihood that an event will occur.
Probability values are from 0 to 1.
5 Slide
Probability as a Numerical Measureof the Likelihood of Occurrence
0 10.5Increasing Likelihood of Occurrence
Probability:
The eventis veryunlikelyto occur.
The occurrenceof the event is
just as likely asit is unlikely.
The eventis almostcertain
to occur.
6 Slide
STATISTICAL EXPERIMENTS
7 Slide
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 Slide
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 3 4 52 6
9 Slide
An Experiment and Its Sample Space
ExperimentToss a coinInspect a partConduct a sales call
Experiment OutcomesHead, tailDefective, non-defectivePurchase, no purchase
10 Slide
Bradley has invested in two stocks, Markley Oil and Collins Mining. Bradley has determined
that thepossible outcomes of these investments three
monthsfrom now are as follows. Investment Gain or Loss
in 3 Months (in $000)Markley Oil Collins Mining
10 5 0-20
8-2
Bradley Investments
An Experiment and Its Sample Space
11 Slide
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 Slide
Bradley Investments can be viewed as a two-step
experiment. It involves two stocks, each with a set of
experimental outcomes.Markley Oil: n1 = 4Collins Mining: n2 = 2Total Number of Experimental Outcomes: n1n2 = (4)(2) = 8
A Counting Rule for Multiple-Step Experiments
Bradley Investments
13 Slide
Tree Diagram
Gain 5
Gain 10
Lose 20Even
Markley Oil(Stage 1)
Collins Mining(Stage 2)
ExperimentalOutcomes
(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 8
Gain 8
Gain 8
Gain 8
Lose 2
Lose 2
Lose 2
Lose 2
Bradley Investments
14 Slide
Combinations enable us to count the number of experimental outcomes when n objects are to be selected from a set of N objects.
Counting Rule for Combinations
CNn
Nn N nn
N
-!
!( )!
Number of Combinations of N Objects Taken n at a Time
where: N! = N(N - 1)(N - 2) . . . (2)(1) n! = n(n - 1)(n - 2) . . . (2)(1) 0! = 1
15 Slide
Number of Permutations of N Objects Taken n at a Time
where: N! = N(N - 1)(N - 2) . . . (2)(1) n! = n(n - 1)(n - 2) . . . (2)(1) 0! = 1
P nNn
NN nn
N
-
! !( )!
Counting Rule for Permutations
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.
16 Slide
Combinations and Permutations
4 Objects: A B C D
12224
!2!4
)!24(!4
)!(!4
2 -
-
nN
NP
A BA CA DB CB DC D
B AC AD AC BD BD C
)!24(!2!4
)!(!!4
2 -
-
nNnNC
6424
2*224
!2!*2!4
A B
A C
A D C D
B D
B C
17 Slide
PRACTICETREE DIAGRAMS, COMBINATIONS, AND PERMUTATIONS
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Practice Tree DiagramA 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.
19 Slide
Combinations
There are five boxes numbered 1 through 5. You pick two boxes.
How many combinations of boxes are there?
Show the combinations.
)!(!!nNn
NC Nn -
20 Slide
Combinations
There are five boxes numbered 1 through 5. You pick two boxes.
How many permutations of boxes are there?
Show the permutations.
)!(!nN
NPNn -
21 Slide