Chap04 basic probability

Statistics for Managers
Using Microsoft® Excel
4th Edition

Chapter 4

Basic Probability

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 4-1

Chapter Goals
After completing this chapter, you should be
able to:
 Explain basic probability concepts and definitions

 Use contingency tables to view a sample space

 Apply common rules of probability

 Compute conditional probabilities

 Determine whether events are statistically

independent
Statistics for Managers Using for conditional probabilities
 Use Bayes’ Theorem

Microsoft Excel, 4e © 2004
Prentice-Hall, Inc. Chap 4-2

Important Terms

 Probability – the chance that an uncertain event
will occur (always between 0 and 1)
 Event – Each possible type of occurrence or
outcome
 Simple Event – an event that can be described
by a single characteristic
 Sample Space – the collection of all possible
events
Statistics for Managers Using

Assessing Probability
 There are three approaches to assessing the probability
of un uncertain event:
1. a priori classical probability
X number of ways the event can occur
probability of occurrence = =
T total number of elementary outcomes

2. empirical classical probability
number of favorable outcomes observed
probability of occurrence =
total number of outcomes observed

3. subjective probability
Statistics for Managers judgment or opinion about the probability of occurrence
an individual Using


Sample Space
The Sample Space is the collection of all
possible events
e.g. All 6 faces of a die:

e.g. All 52 cards of a bridge deck:


Events
 Simple event
 An outcome from a sample space with one
characteristic
 e.g., A red card from a deck of cards
 Complement of an event A (denoted A’)
 All outcomes that are not part of event A
 e.g., All cards that are not diamonds
 Joint event
 Involves two or more characteristics simultaneously
Statisticsfor Managers that is also red from a deck of cards
e.g., An ace Using

Visualizing Events
 Contingency Tables
Ace Not Ace Total

Black 2 24 26
Red 2 24 26

Total 4 48 52

 Tree Diagrams Sample
Ac e 2 Space
Sample
ar d
Space
Black C No t a n A c e 24
Full Deck
of 52 Cards Ace
Statistics for Managers Red Ca
Using 2
rd
Microsoft Excel, 4e © 2004 No t a n
Ace 24

Mutually Exclusive Events
 Mutually exclusive events
 Events that cannot occur together

example:

A = queen of diamonds; B = queen of clubs

 Events A and B are mutually exclusive


Collectively Exhaustive Events
 Collectively exhaustive events
 One of the events must occur
 The set of events covers the entire sample space

example:
A = aces; B = black cards;
C = diamonds; D = hearts
 Events A, B, C and D are collectively exhaustive
(but not mutually exclusive – an ace may also be
a heart)
Statistics for Managersand D are collectively exhaustive and
 Events B, C
Using
also mutually exclusive

Probability
 Probability is the numerical measure
of the likelihood that an event will 1 Certain

occur
 The probability of any event must be
between 0 and 1, inclusively
0 ≤ P(A) ≤ 1 For any event A .5
 The sum of the probabilities of all
mutually exclusive and collectively
exhaustive events is 1
P(A) + P(B) + P(C) = 1
Statistics for Managers Using 0 Impossible
If A, B, and C are mutually exclusive and
collectively exhaustive

Computing Joint and
Marginal Probabilities

 The probability of a joint event, A and B:
number of outcomes satisfying A and B
P( A and B) =
total number of elementary outcomes

 Computing a marginal (or simple) probability:

P(A) = P(A and B1 ) + P(A and B 2 ) +  + P(A and Bk )

Statistics for Managers2,Usingare k mutually exclusive and collectively
Where B1, B …, Bk


exhaustive events

Joint Probability Example

P(Red and Ace)
number of cards that are red and ace 2
= =
total number of cards 52

Color
Type Red Black Total
Ace 2 2 4
Non-Ace 24 24 48
Statistics forTotal 26
Managers Using 26 52

Marginal Probability Example

P(Ace)
2 2 4
= P( Ace and Re d) + P( Ace and Black ) = + =
52 52 52

Color
Ace 2 2 4
Non-Ace 24 24 48
Statistics forTotal 26
Managers Using 26 52

Joint Probabilities Using
Contingency Table

Event
Event B1 B2 Total
A1 P(A1 and B1) P(A1 and B2) P(A1)

A2 P(A2 and B1) P(A2 and B2) P(A2)

Total P(B1) P(B2) 1

Joint for Managers Using Marginal (Simple) Probabilities
Statistics Probabilities

General Addition Rule

General Addition Rule:
P(A or B) = P(A) + P(B) - P(A and B)

If A and B are mutually exclusive, then
P(A and B) = 0, so the rule can be simplified:

P(A or B) = P(A) + P(B)
For mutually exclusive events A and B

General Addition Rule Example

P(Red or Ace) = P(Red) +P(Ace) - P(Red and Ace)

= 26/52 + 4/52 - 2/52 = 28/52
Don’t count
the two red
Color aces twice!
Ace 2 2 4
Non-Ace 24 24 48
Total 26 26 52

Computing Conditional
Probabilities
 A conditional probability is the probability of one
event, given that another event has occurred:
P(A and B) The conditional
P(A | B) = probability of A given
P(B) that B has occurred

P(A and B) The conditional
P(B | A) = probability of B given
P(A) that A has occurred

Where P(A and B) = joint probability of A and B
Statistics for Managers Using probability of A
P(A) = marginal
P(B) = marginal probability of B

Conditional Probability Example

 Of the cars on a used car lot, 70% have air
conditioning (AC) and 40% have a CD player
(CD). 20% of the cars have both.

 What is the probability that a car has a CD
player, given that it has AC ?

i.e., we want to find P(CD | AC)


(continued)
 Of the cars on a used car lot, 70% have air conditioning
(AC) and 40% have a CD player (CD).
20% of the cars have both.
CD No CD Total
AC .2 .5 .7
No AC .2 .1 .3
Total .4 .6 1.0

P(CD and AC) .2
P(CD | AC) = = = .2857
P(AC) .7

(continued)
 Given AC, we only consider the top row (70% of the cars). Of these,
20% have a CD player. 20% of 70% is about 28.57%.

CD No CD Total
AC .2 .5 .7
No AC .2 .1 .3
Total .4 .6 1.0

P(CD and AC) .2
P(CD | AC) = = = .2857
P(AC)
Statistics for Managers Using .7

Using Decision Trees
.2

Given AC or C D .7 P(AC and CD) = .2
Has
no AC: 7
C )= . D oe
P(A have
s no
t .5 P(AC and CD’) = .5
C CD
a sA
H .7
All
Cars
Do .2
e
hav s not
eA
C P(A C D .3 P(AC’ and CD) = .2
C’) Has
= .3
D
Statistics for Managers Usingh oes not
CD .1 P(AC’ and CD’) = .1
ave
Prentice-Hall, Inc. .3 Chap 4-21

Using Decision Trees
(continued)
.2
C
.4 P(CD and AC) = .2
Given CD or Has A
no CD: 4
D )= . D oe
P(C have
s no
t .2 P(CD and AC’) = .2
C D AC
H as .4
All
Cars
Do .5
e
hav s not .6
eC AC P(CD’ and AC) = .5
D P(C Has
D’)
= .6
D
Statistics for Managers Usingh oes not
AC .1 P(CD’ and AC’) = .1
ave
Prentice-Hall, Inc. .6 Chap 4-22

Statistical Independence
 Two events are independent if and only
if:

P(A | B) = P(A)
 Events A and B are independent when the probability
of one event is not affected by the other event


Multiplication Rules

 Multiplication rule for two events A and B:
P(A and B) = P(A | B) P(B)

Note: If A and B are independent, then P(A | B) = P(A)
and the multiplication rule simplifies to

P(A and B) = P(A) P(B)

Marginal Probability

 Marginal probability for event A:

P(A) = P(A | B1 ) P(B1 ) + P(A | B 2 ) P(B 2 ) +  + P(A | Bk ) P(Bk )

 Where B1, B2, …, Bk are k mutually exclusive and
collectively exhaustive events


Bayes’ Theorem Example

 A drilling company has estimated a 40%
chance of striking oil for their new well.
 A detailed test has been scheduled for more
information. Historically, 60% of successful
wells have had detailed tests, and 20% of
unsuccessful wells have had detailed tests.
 Given that this well has been scheduled for a
detailed test, what is the probability
Statistics for Managers Usingsuccessful?
that the well will be

(continued)

 Let S = successful well
U = unsuccessful well
 P(S) = .4 , P(U) = .6 (prior probabilities)
 Define the detailed test event as D
 Conditional probabilities:
P(D|S) = .6 P(D|U) = .2
 Goal is to find P(S|D)

(continued)

Apply Bayes’ Theorem:
P(D | S)P(S)
P(S | D) =
P(D | S)P(S) + P(D | U)P(U)
(.6)(.4)
=
(.6)(.4) + (.2)(.6)
.24
= = .667
.24 + .12

Statistics forrevised probability of success, given that this well
So the Managers Using
Microsoft Excel, 4e © 2004 for a detailed test, is .667
has been scheduled

(continued)

 Given the detailed test, the revised probability
of a successful well has risen to .667 from the
original estimate of .4

Event Prior Conditional Joint Revised
Prob. Prob. Prob. Prob.
S (successful) .4 .6 .4*.6 = .24 .24/.36 = .667
U (unsuccessful) .6 .2 .6*.2 = .12 .12/.36 = .333

Statistics for Managers Using Sum = .36

Chapter Summary
 Discussed basic probability concepts
 Sample spaces and events, contingency tables, simple
probability, and joint probability
 Examined basic probability rules
 General addition rule, addition rule for mutually exclusive events,
rule for collectively exhaustive events
 Defined conditional probability
 Statistical independence, marginal probability, decision trees,
and the multiplication rule
 Discussed Bayes’ theorem

Chap04 basic probability

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