2. Some Basic Probability Concepts

2014.3.13

2.1 Introduction

• Probability theory

– Set

– Probability measure P(∙) : A -> [0,1]

2.2 Two views of probability: objective & subjective

• Objective probability

1) 고전적이거나 사전 확률 (a priori)

(이론적이고 추상적인 개념

예, 주사위의 한 면 1/6)

(theoretical and abstract, dice example)

(N mutually exclusive events with equal

probability, m events)

2) 상대도수 또는 사후확률 (a posteriori)

(실험적이고 실제적, 해보자)

(empirical, n:#trials, m:#events)

N

mEP )(

n

mEP )(

2.3 elementary properties of probability

1)

2)

3)

0)( iEP

1)()()( 21 nEPEPEP

)()()( jiji EPEPEorEP

1 2, , , nE E E mutually exclusive exhaustive events

2.4 Using set theory

• 요소(element) 혹은 항(member) 1) 집합의 모든 요소를 열거

2) 집합을 구성하는 요소의 형태에 따라 열거 • 단위집합(unit set): 단지 하나의 요소로 구성

(set of only one element) • 공집합(empty set, null set) • 전체집합(universal set) U • 부분집합(subset) • 공집합은 모든 집합의 부분집합 (Empty set

is a subset of all sets) • 동일한 집합 (Two sets are identical if and

only if all the elements are the same)

• 합집합(union)

1) 결합(conjoint)

2) 분리(disjoint)

• 교집합(intersection)

• 여집합(complement)

Venn diagram

< 보기 2.4.1 >

75)( 41 ABn

60330513120)( 22 ABn

13813851766)( 4 An

2.5 permutation and combination

• Factorials :

< 보기 2.5.1 >

서로 다른 미생물 배양하는 배지용기 4개.

선반 위에 한 줄로 놓는다면?

Put (A,B,C,D) in a row

4! = 4*3*2*1 = 24

)!1(1)3)(2)(1(! nnnnnnn

4! 4 3 2 1 24

순열(Permutations) –Order matters

< 보기 2.5.1 > choose 2 out of 4

12)!24(

!424

P

)!(

!)1()2)(1(

rn

nrnnnnPrn

< 보기 2.5.2 >

5 rooms (A,B,C,D,E) in a clinic

Assign 5 nurses to the rooms

• 조합(combination) – Order does not matter

120)!55(

!555

P

! nn r rP r

!

! !( )!

n n rr

P n

r r n r

42

4!6

2!2!

choose 2 out of 4(A,B,C,D)

<보기 2.5.3>

Choose 6 patients from 10 pts

106

10!210

6!4!

• Permutation with groups of identical objects

<보기 2.5.4>

5 nurses (A,B,C,D,E) paint 5 rooms 개 진료실에 페인트칠. 2 nurses with white paint, 2 with yellow, 1 with green. How many ways?

1 2, , ,

1 2

!

! ! !kn n n n

k

nP

n n n

1 2, , , 1 2! ( ) ! ! !kn n n n kn P n n n

5 2,2,1

5!30

2!2!1!P

<보기 2.5.5>

2 white, 2 green, 2 yellow vegetables in a kitchen. How many ways to display them in a row?

6 2,2,2

6!90

2!2!2!P

2.6 Calculating probability

<보기 2.6.1>

table 2.4.1, select one person. Probability that age of him/her is less than or equal to 25

• 조건부 확률(conditional probability)

11

( ) 260( ) .15

( ) 1766

n AP A

n U

( )( ) , ( ) 0

( )

P A BP A B P B

P B

표 2.4.1 Probability of selecting a doctor?

Probability of selecting a doctor from the groups of age > 35 (A4 )?

11

( ) 105( ) .06

( ) 1766

n BP B

n U

1 4

1 4 1 41 4

44 4

( )

( ) ( ) 75( )( ) .19

( )( ) ( ) 385

( )

n B A

P B A n B An UP B A

n AP A n A

n U

• 주변확률(marginal probability)

P of selecting a person with age > 35?

• 가산법칙(addition rule)

• 승산법칙(multiplication rule)

• 독립(independence)

4

385( ) .2180

1766P A

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

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

( ) ( )P A B P A

<보기 2.6.2>

60 girls 40 boys in a class. 24 girls, 16 boys are wearing glasses. P ( select a boy, he wears glasses)?

(24+16)/(60+40)=40/100=

So E and B are independent.

16100

40100

( )( ) .4

( )

P E BP E B

P B

( ) ( )P E P E B

40 40100 100

( ) ( ) ( )

( ) ( ) .16

P E B P B P E B

P B P E

• A, B are independent and

• are mutually exclusive

<보기 2.6.3>

750 pts are admitted individually out of 1200pts

( ) 0, ( ) 0P A P B

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

( ) 1 ( )P A P A

( ), ( )P A P A

7501200

4501200

( ) .625

( ) .375

( ) 1 ( ) 1 .625 .375

P A

P A

P A P A

• About statistical independence

• SEX, smoking

• 48% male, 55% smoker : marginal probability

• If they are independent, we know joint probability.

P(Male and Smoker) =P(Male)*P(Smoker)=0.48*0.55=0.26

• From Marginal prob Male Female total

Smoker

.55

Non-smoker

1-.55=.45

total .48 1-.48=.52

1

Male Female total

Smoker

.55*.48=.26

.55*.52 or .55-.26

.55

Non-smoker

.45*.48 or .48-.26

.45*.52 1-.55=.45

total .48 1-.48=.52 1

If they are independent, then

• If two variables are independent, than we know their joint probability with their marginal probabilities only.

• But, we never know their joint probabilities from their marginal probabilities. -> We need information on the combinations of their values.

• We need to make plans on the combination of the variables in the study planning stage.

• We will learn more on this issue at chap 10.

• Independent case

RR=(%SMK for Male)

/(%SMK for Female)

=.55/.55=1

Male

(%Smk)

Female total

Smoker 26

(55)

29

(55)

55

Non-smoker

22 23 45

total 48 52 100

Male

(%Smk)

Female

(%Smk)

total

Smoker 39

(81)

16

(31)

55

Non-smoker

9 36 45

total 48 52 100

Dependent case

RR=(%SMK for Male)

/(%SMK for Female)

=.81/.31=2.61

homework

• 2.4.1 2.6.1 2.6.4

• 종합문제 11 14