2. some basic probability conceptshosting03.snu.ac.kr/~hokim/int/2014/chap2.pdf2.6 calculating...
TRANSCRIPT
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