Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-1 Chapter 4 Basic Probability Business Statistics: A First Course 5 th Edition

Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-1

Chapter 4

Basic Probability

Business Statistics:A First Course

5th Edition


Learning Objectives

In this chapter, you learn:

Basic probability concepts（概率的基本概念） Conditional probability （条件概率） To use Bayes’ Theorem to revise probabilities（利用贝叶斯定理来修正概率）


Basic Probability Concepts

Probability – the chance that an uncertain event will occur (always between 0 and 1)（一个不确定事件发生的可能性称为概率）

Impossible Event – an event that has no chance of occurring (probability = 0)（如果一个事件发生概率为零，则称为不可能事件）

Certain Event – an event that is sure to occur (probability = 1)（相反，如果事件发生概率为 1，则为必然事件）


Assessing Probability

There are three approaches to assessing the probability of an uncertain event（三种评价事件发生概率的方法） :1. a priori -- based on prior knowledge of the process（先验概率，基于预先掌握的知识）

2. empirical probability（经验概率，基于观察到的数据）

3. subjective probability（主观概率，每个人的主观可能不一样）

outcomeselementaryofnumbertotal

occurcaneventthewaysofnumber

T

X

based on a combination of an individual’s past experience, personal opinion, and analysis of a particular situation 依赖于每个人不同的经验、观点和对特定情形分析的结合


occurcaneventthewaysofnumber

Assuming all outcomes are equally likely要假定各种结果出现的可能性是相同的

probability of occurrence

probability of occurrence


Example of a priori probability

Find the probability of selecting a face card (Jack, Queen, or King) from a standard deck of 52 cards. （ 52张牌中一次抽到人牌的概率）

cards ofnumber total

cards face ofnumber Card Face ofy Probabilit

T

X

13

3

cards total52

cards face 12

T

X


Example of empirical probability

Taking Stats Not Taking Stats

Total

Male 84 145 229

Female 76 134 210

Total 160 279 439

Find the probability of selecting a male taking statistics from the population described in the following table（依照下表描述的总体，计算一次抽到选择统计的男生的概率） :

191.0439

84

people ofnumber total

stats takingmales ofnumber Probability of male taking stats


Events

Each possible outcome of a variable is an event.（变量的每一个可能结果称为事件）

Simple event（简单事件） An event described by a single characteristic（对应单一特征） e.g., A red card from a deck of cards

Joint event（联合事件） An event described by two or more characteristics（两个或者两个以上特征的时间）

e.g., An ace that is also red from a deck of cards Complement of an event A (denoted A’)（事件 A的补）

All events that are not part of event A（所有不包含事件 A的事件集合）

e.g., All cards that are not diamonds


Sample Space（样本空间）

The Sample Space is the collection of all possible events（所有可能事件的集合称为样本空间）

e.g. All 6 faces of a die:

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


Visualizing Events

Contingency Tables（列联表）

Decision Trees（决策树）

Red 2 24 26

Black 2 24 26

Total 4 48 52

Ace Not Ace Total

Full Deck of 52 Cards

Red Card

Black Card

Not an Ace

Ace

Ace

Not an Ace Sample Space

Sample Space

2

24

2

24


Visualizing Events

Venn Diagrams（韦恩图示法） Let A = aces Let B = red cards

A

B

A ∩ B = ace and red

A U B = ace or red


DefinitionsSimple vs. Joint Probability

Simple Probability refers to the probability of a simple event.（简单事件发生的概率称为简单概率） ex. P(King) ex. P(Spade黑桃 )

Joint Probability refers to the probability of an occurrence of two or more events (joint event).（联合事件发生的概率称为联合概率） ex. P(King and Spade)


Mutually Exclusive Events（互斥事件）

Mutually exclusive events Events that cannot occur simultaneously（不能同时发生的事件）

Example: Drawing one card from a deck of cards

A = queen of diamonds方块 ; B = queen of clubs梅花

Events A and B are mutually exclusive


Collectively Exhaustive Events（完备事件集）

Collectively exhaustive events One of the events must occur（事件集中某个事件一定发生）

The set of events covers the entire sample space（整个事件集覆盖整个样本空间）

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)（完备事件集中事件不一定互斥）

Events B, C and D are collectively exhaustive and also mutually exclusive


Computing Joint and Marginal Probabilities联合概率及边际概率的计算

The probability of a joint event, A and B:

Computing a marginal (or simple) probability:

Where B1, B2, …, Bk are k mutually exclusive and collectively exhaustive events（互斥且完备事件集）


BandAsatisfyingoutcomesofnumber)BandA(P

)BdanP(A)BandP(A)BandP(AP(A) k21


Joint Probability Example

P(Red and Ace)

BlackColor

Type Red Total

Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

52

2

cards of number total

ace and red are that cards of number


Marginal Probability Example

P(Ace)

BlackColor

Type Red Total

Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

52

4

52

2

52

2)BlackandAce(P)dReandAce(P


P(A1 and B2) P(A1)

TotalEvent

Marginal & Joint Probabilities In A Contingency Table

P(A2 and B1)

P(A1 and B1)

Event

Total 1

Joint Probabilities Marginal (Simple) Probabilities

A1

A2

B1 B2

P(B1) P(B2)

P(A2 and B2) P(A2)


Probability Summary So Far

Probability is the numerical measure of the likelihood that an event will occur（概率是刻画一个事件发生可能性的数字测度）

The probability of any event must be between 0 and 1, inclusively（任何事件发生概率值都在 0 和 1之间）

The sum of the probabilities of all mutually exclusive and collectively exhaustive events is 1（互斥且完备事件集的概率之和为 1）

Certain

Impossible

0.5

1

0

0 ≤ P(A) ≤ 1 For any event A

1P(C)P(B)P(A) If A, B, and C are mutually exclusive and collectively exhaustive


General Addition Rule（一般加法原则）

P(A or B) = P(A) + P(B) - P(A and B)

General Addition Rule:

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 = 28/52Don’t count the two red aces twice!

BlackColor

Type Red Total

Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52


Computing Conditional Probabilities条件概率的计算

A conditional probability is the probability of one event, given that another event has occurred:一个事件在给定另一事件发生下发生的概率

P(B)

B)andP(AB)|P(A

P(A)

B)andP(AA)|P(B

Where P(A and B) = joint probability of A and B （ A 、 B的联合概率） P(A) = marginal or simple probability of A （ A的边际概率）

P(B) = marginal or simple probability of B （ B的边际概率）

The conditional probability of A given that B has occurred

The conditional probability of B given that A has occurred


What is the probability that a car has a CD player, given that it has AC ?（问在那些有空调的车中，拥有 CD机的概率？）

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

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.在某个二手车市场， 70%的车有空调， 40%的车有 CD机， 20%的车这两样东西都有



No CDCD Total

AC 0.2 0.5 0.7

No AC 0.2 0.1 0.3

Total 0.4 0.6 1.0

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.

0.28570.7

0.2

P(AC)

AC)andP(CDAC)|P(CD

(continued)



No CDCD Total

AC 0.2 0.5 0.7

No AC 0.2 0.1 0.3

Total 0.4 0.6 1.0

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%.

0.28570.7

0.2

P(AC)

AC)andP(CDAC)|P(CD

(continued)


Using Decision Trees

Has AC

Does not have AC

Has CD

Does not have CD

Has CD

Does not have CD

P(AC)= 0.7

P(AC’)= 0.3

P(AC and CD) = 0.2

P(AC and CD’) = 0.5

P(AC’ and CD’) = 0.1

P(AC’ and CD) = 0.2

7.

5.

3.

2.

3.

1.

AllCars

7.

2.

Given AC or no AC:

ConditionalProbabilities


Using Decision Trees

Has CD

Does not have CD

Has AC

Does not have AC

Has AC

Does not have AC

P(CD)= 0.4

P(CD’)= 0.6

P(CD and AC) = 0.2

P(CD and AC’) = 0.2

P(CD’ and AC’) = 0.1

P(CD’ and AC) = 0.5

4.

2.

6.

5.

6.

1.

AllCars

4.

2.

Given CD or no CD:

(continued)

ConditionalProbabilities


Independence（独立性） Two events are independent if and only

if:

Events A and B are independent when the probability of one event is not affected by the fact that the other event has occurred（两个事件 A 、 B是相互独立的，如果 A是否发生不受 B已经发生的影响）

P(A)B)|P(A


Multiplication Rules（乘法法则）

Multiplication rule for two events A and B:

P(B)B)|P(AB)andP(A

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

P(B)P(A)B)andP(A


Marginal Probability

Marginal probability for event A:

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

化边际概率的计算为多个条件概率乘积的和

)P(B)B|P(A)P(B)B|P(A)P(B)B|P(A P(A) kk2211


Bayes’ Theorem

Bayes’ Theorem is used to revise previously calculated probabilities based on new information.（基于新的信息修正原先已经计算的概率）

Developed by Thomas Bayes in the 18th Century.（由托马斯 -贝叶斯发展于 18世纪）

It is an extension of conditional probability.（条件概率的一个推广）


Bayes’ Theorem（贝叶斯定理）

where:

Bi = ith event of k mutually exclusive and collectively

exhaustive events （ Bi ， k个互斥且完备的事件）

A = new event that might impact P(Bi)

))P(BB|P(A))P(BB|P(A))P(BB|P(A

))P(BB|P(AA)|P(B

k k 2 2 1 1

i i i


Bayes’ Theorem Example

A drilling company has estimated a 40% chance of striking oil for their new well. 一家石油钻探公司新钻井出油成功率大概在 40%，可以看成先验成功率

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. 详细的钻前勘探能带来更多信息。历史上成功出油的钻井中 60%做过详细的钻前勘探，没有出油的钻井中 20%做过详细的钻前勘探

Given that this well has been scheduled for a detailed test, what is the probability that the well will be successful?（假定该次钻探前，已经做了详细的钻前勘探，请问该井成功出油的概率？）


Let S = successful well

U = unsuccessful well P(S) = 0.4 , P(U) = 0.6 (prior probabilities)

Define the detailed test event as D

Conditional probabilities:

P(D|S) = 0.6 P(D|U) = 0.2

Goal is to find P(S|D)

Bayes’ Theorem Example(continued)


0.6670.120.24

0.24

(0.2)(0.6)(0.6)(0.4)

(0.6)(0.4)

U)P(U)|P(DS)P(S)|P(D

S)P(S)|P(DD)|P(S

Bayes’ Theorem Example(continued)

Apply Bayes’ Theorem:

So the revised probability of success, given that this well has been scheduled for a detailed test, is 0.667


Given the detailed test, the revised probability of a successful well has risen to 0.667 from the original estimate of 0.4

Bayes’ Theorem Example

EventPrior

Prob.

Conditional Prob.

Joint

Prob.

Revised

Prob.

S (successful) 0.4 0.6 (0.4)(0.6) = 0.24 0.24/0.36 = 0.667

U (unsuccessful) 0.6 0.2 (0.6)(0.2) = 0.12 0.12/0.36 = 0.333

Sum = 0.36

(continued)


Chapter Summary

Discussed basic probability concepts Sample spaces and events, contingency tables, Venn diagrams,

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

Documents

Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 4-1 Chapter 4 Basic Probability Business Statistics: A First Course 5 th Edition