Transcript P(B 2 ) - Webster in china

Business Statistics:
A First Course
5th Edition
Chapter 4
Basic Probability
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-1
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）
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-2
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）
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-3
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）
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 
X
number of ways the event can occur

T
total number of elementary outcomes
2. empirical probability（经验概率，基于观察到的数据）
probability of occurrence 
number of ways the event can occur
total number of elementary outcomes
3. subjective probability（主观概率，每个人的主观可能不一样）
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
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）
X
number of face cards
Probabilit y of Face Card 

T
total number of cards
X
12 face cards
3


T
52 total cards 13
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-5
Example of empirical probability
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）:
Taking Stats
Not Taking
Stats
Total
Male
84
145
229
Female
76
134
210
160
279
439
Total
Probability of male taking stats 
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
numberof males taking stats 84

 0.191
totalnumberof people
439
Chap 4-6
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）



Joint event（联合事件）



An event described by a single characteristic（对应单一特征）
e.g., A red card from a deck of cards
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
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-7
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:
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-8
Visualizing Events

Contingency Tables（列联表）
Ace

Sample
Space
Not Ace
Total
Black
2
24
26
Red
2
24
26
Total
4
48
52
Decision Trees（决策树）
2
Sample
Space
24
Full Deck
of 52 Cards
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
2
24
Chap 4-9
Visualizing Events

Venn Diagrams（韦恩图示法）

Let A = aces

Let B = red cards
A ∩ B = ace and red
A
A U B = ace or red
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
B
Chap 4-10
Definitions
Simple 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)
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-11
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
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-12
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
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-13
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:
number of outcomes satisfyingA 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 B2 )    P(A and Bk )

Where B1, B2, …, Bk are k mutually exclusive and collectively
exhaustive events（互斥且完备事件集）
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-14
Joint Probability Example
P(Red and Ace)

number of cards that are red and ace
2

total number of cards
52
Type
Color
Red
Black
Total
Ace
2
2
4
Non-Ace
24
24
48
Total
26
26
52
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-15
Marginal Probability Example
P(Ace)
 P( Ace and Re d)  P( Ace and Black ) 
Type
2
2
4


52 52 52
Color
Red
Black
Total
Ace
2
2
4
Non-Ace
24
24
48
Total
26
26
52
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-16
Marginal & Joint Probabilities In A
Contingency Table
Event
B1
Event
B2
Total
A1
P(A1 and B1) P(A1 and B2)
A2
P(A2 and B1) P(A2 and B2) P(A2)
Total
P(B1)
Joint Probabilities
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
P(B2)
P(A1)
1
Marginal (Simple) Probabilities
Chap 4-17
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）
Certain
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）
0 ≤ P(A) ≤ 1 For any event A

1
0.5
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）
P(A)  P(B)  P(C)  1
If A, B, and C are mutually exclusive and
collectively exhaustive
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
0
Impossible
Chap 4-18
General Addition Rule（一般加法原则）
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
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-19
General Addition Rule Example
P(Red or Ace) = P(Red) +P(Ace) - P(Red and Ace)
= 26/52 + 4/52 - 2/52 = 28/52
Type
Color
Red
Black
Total
Ace
2
2
4
Non-Ace
24
24
48
Total
26
26
52
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Don’t count
the two red
aces twice!
Chap 4-20
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(A and B)
P(A | B) 
P(B)
The conditional
probability of A given
that B has occurred
P(A and B)
P(B | A) 
P(A)
The conditional
probability of B given
that A has occurred
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）
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-21
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
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)
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-22
Conditional Probability Example
(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
0.2
0.5
0.7
No AC
0.2
0.1
0.3
Total
0.4
0.6
1.0
P(CD and AC) 0.2
P(CD | AC) 

 0.2857
P(AC)
0.7
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-23
Conditional Probability Example
(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
0.2
0.5
0.7
No AC
0.2
0.1
0.3
Total
0.4
0.6
1.0
P(CD and AC) 0.2
P(CD | AC) 

 0.2857
P(AC)
0.7
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-24
Using Decision Trees
.2
.7
Given AC or
no AC:
.5
.7
All
Cars
P(AC and CD) = 0.2
P(AC and CD’) = 0.5
Conditional
Probabilities
.2
.3
.1
.3
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
P(AC’ and CD) = 0.2
P(AC’ and CD’) = 0.1
Chap 4-25
Using Decision Trees
.2
.4
Given CD or
no CD:
.2
.4
All
Cars
(continued)
P(CD and AC) = 0.2
P(CD and AC’) = 0.2
Conditional
Probabilities
.5
.6
.1
.6
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
P(CD’ and AC) = 0.5
P(CD’ and AC’) = 0.1
Chap 4-26
Independence（独立性）

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 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）
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-27
Multiplication Rules（乘法法则）

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)
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-28
Marginal Probability

Marginal probability for event A:
P(A)  P(A | B1)P(B1)  P(A | B2 )P(B 2 )    P(A | Bk )P(Bk )


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
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-29
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）
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-30
Bayes’ Theorem（贝叶斯定理）
P(A| B i )P(Bi )
P(Bi | A) 
P(A| B 1 )P(B1 )  P(A| B 2 )P(B2 )      P(A| B k )P(Bk )

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)
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-31
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？）
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-32
Bayes’ Theorem Example
(continued)

Let S = successful well
U = unsuccessful well

P(S) = 0.4 , P(U) = 0.6

Define the detailed test event as D

Conditional probabilities:
P(D|S) = 0.6

(prior probabilities)
P(D|U) = 0.2
Goal is to find P(S|D)
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-33
Bayes’ Theorem Example
(continued)
Apply Bayes’ Theorem:
P(D | S)P(S)
P(S | D) 
P(D | S)P(S)  P(D | U)P(U)
(0.6)(0.4)

(0.6)(0.4) (0.2)(0.6)
0.24

 0.667
0.24  0.12
So the revised probability of success, given that this well
has been scheduled for a detailed test, is 0.667
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-34
Bayes’ Theorem Example
(continued)

Given the detailed test, the revised probability
of a successful well has risen to 0.667 from
the original estimate of 0.4
Event
Prior
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
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-35
Chapter Summary

Discussed basic probability concepts


Examined basic probability rules


General addition rule, addition rule for mutually exclusive events,
rule for collectively exhaustive events
Defined conditional probability


Sample spaces and events, contingency tables, Venn diagrams,
simple probability, and joint probability
Statistical independence, marginal probability, decision trees,
and the multiplication rule
Discussed Bayes’ theorem
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 4-36