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.???????
???? ??????? ?????????3:?????????
Collection Presentation Analysis
Stat 500 AS507
:???????? ?????????
1.( ???????????? ???????????Probability Theory.)
2.( ??????????? ????????????? ??????Probability Distribution.)
3.( ???????? ???????????Sampling Distribution.)
4..???????? ??????????????
Chapter 1: Probability
:????????? ???? ??????? ??????
??????????????? ?????????????(Random Experiment.)(1)
)??????? ?(??? ????????? ?????????????(2)
??????????????? ???????????????????????? ????????????? ????? ?????????? ???? ???????? ????? ????????????????????????????? ?????? ???? ?????? ???? ????????
????????????? ??????????? ????? ??????? ?????? ????????? ????? ???????????????????? ??????????? ????? ??????????????? ????? ???????? ????? ???????? ?(??????
.???????? ?????????? ??????? ???????? ???? ????????????? ????? ??????? ?????? ???? ????? ?)?????????? ?????????????
???????? ??????????(Sample space:)????????????????????? ???????????????? ????? ???????????????????????????????????????????????????????????????
(Domain?)????? ?????? ??????(1?2?3?4?5?6??????? )????????????????(?????????????):???????? ?????? ??????? ?
S = {H, T} S = {1, 2, 3, 4, 5, 6} S = {D, N} D = Defective
??????????:???????? ?????????? ???
( ?????? ???? ???????Continuous)( ???????Discrete)
( ?????? ?????? ????? ????????? ??????? ?????uncountable)
??????? ???? ????? ??????? )??????? ???? ?(?????? ?????? ?????? ????
( ??????? ????[ ])
)?????????????? ???? ????????????? ?(???? ?????? ??????? ??????? ????? ?????
( ???????Finite)( ??????? ?????Infinite)
(1????? ?????? )???:??????????????? ???????? ???( ??????? ??????die)?( ?????? ?????coin)??????????? ??????? ?????????????? ?
(2??????? ????? ????? ).?????????? ???????? ??????? ???????? ??????? ?????? ????? ??????? ????? ?????????? ????????????? ??????????? ????? ????? ???????? ?????????

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???????? ???????? ???????? ?????????? ?????????????:????????
S = {x : 0 ≤ x ≤ 100, x R}
" ?????? ???? ???? ???????? ?????????? :????????Order.???????? ?????????? ?????? ????????? ????? ?????? ????? "
( ????????????? ???????Sub Set:)????????????? ???????? ????? ??????????? ????????? ????????? ?????????? ????????? ?????? ???????? ???? ???????? ????
:???????? ????????? ???? ??????????
=2Order
, Order = ???????? ?????????? ??????? ?????????????? ???? ??????? ????
:???? ????????? ?????????? ????? ????????? ?????? ????? ???? ????????????? ??????????? :??????{H}, {T}, {H, T}, {} or
???????(The Event:)??? ???????????? ?????????? ???? ??????????? ????? ????(3)
:??????? ????? ???? ??????????? ?????? ???????
( ???????? ?????Impossible) ?????( ?????????Probable) ( ???????? ?????Sure)
P=0 0 < P < 1 P=1
:?????? ????????? ????? ????????? ???? ???????? ????? ?????
( ?????? ???????Mutually Exclusive Event)( ????????? ???????The Independent Events)
?????? ?????? ??????????? ????? :??????? ?????? ?????? ????? ??????
??????? ???? ???????
?????? ??????????????? ?????? ???????? ????? ?????? ???? ????????
.?????????????? ?????(4)
???????? ???????(Compliment Event).
( ???????? ???? ???????Ac
( ???? )A’???? ?????????????? ???? ???????? ??????? ?????? ????? ?????????? ????????? ???????? ??????? ???????? ?)
???????? ?????????? ???????? ???????? ???????.(5)
P (A) + P (A’) = 1
(3???????? ?????????? ???? ????????????? ????? ??????????? ????? ????).????? ????????? ???? )?(???? ????????? ?????????? ????? ?????? ???????
(4)????? ????????? ?????? ?????? ??????? ????????? ?????? ???? ??????? ?????????? ????????? ?????? ?????? ???????? ?????? ???? ????? ?????????? ???????? ????? ??????????? ????????? ????????? ?????
?????? ????????? :????????? ??????????? ???????????? ??????? ????? ???????????????????? ????? ????????? ?????????? ????? ??????? ????? ??????????? ????????? ?????????? ????? ??????? ????? ????????
.????????? ????? ???? ????????? ?????? ????? ?????????? ????? ????????? ????????? ????????? ????? ???? ???????? ?????? ????
(5):????????????? ?????? ???? ???????? ????????????? ??????????? ??????3= ?????? ??????? ?????? ?????? ???????????? ?3?????? ????P=3/36?
p(A)=1-p(Ac) = 1- (3/36) = 33/36

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:????????? ???????? ????????? ????? ???????
?????? ??????????????? ???????
A B =A B = , A B = S
.??????? ???????? ?????????? ??????? ????? ?????????? ????????? ???? ??????? ???????? ????
?????????? ??????:
????????????? ?????????????????? ????(Mathematical prob.)?????????????? ??????????(Empirical prob.)
?????????? ???????? ?????????? ????? ??????? ???????????? ?????????????? ????????????? ????? ????????????????????????????
???????? ??????????? ????? ?????????????? ?????????? ????? ??????? ?????????? ????????????????? ????????????? ??????????? ????????? ???????? ???? ?????? ????
.????????????? ?????????? ?????????????? ??????????? ?????(6)
???????? ?????????? ????? ????????? "????????????? ?"????????? ???? ????? ??????? ?????
.??????????
?????????? ?????? ???????:?????????
1.????????????? ?????????????? ??????? ???????????? ??????? ???????( ????????n.)
2.?????? ??????( ???A????? )??????????? ?????( = ???????? ?????? ????????? ?????????? ????m.)
3.??????? ???????? ??????(A):??????????
:??????????? ?????? ???????S = {1, 2, 3, 4, 5, 6}
1.????? ???????? ?????????????? ?????E1 = {2, 4, 6}, P1 = 3/6 = ?
2.???????? ???????????? ?????? ?????4E2 = {5, 6}, P2 = 2/6 = 1/3
3.???????????? ????? ????? ????? ????????1E3 = {}, P3 = 0/6 = 0
??????????:?????????????????? ???????? ????? ?????? ????? ???????????????? ??????? ????? ????? ?????????? ?????? ???? ?????????? ????
0 ≤ P (E) ≤ 1
(6????????? ?????? ????????? ????? ?????? ????? ????? )111?????????? ???? ????? ?%51?????? ?%3?????????? ??????? ???????66??? %33????????????? ?????????? ????? %
51%???? ?????????? ???????? ???????????? ????????????? ????? ?.????????? ???? ????????????

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:????????????????? ????? ?????? ???????
1.?????????? ??????? ????? ???????? ????????E1 ={11, 22, 33, 44, 55, 66}, P1 = 6/36
2.????????? ??????? ????? ???????? ????????11E2 ={46, 55, 64}, P2 = 3/36
3.???? ????? ????????? ??????? ?????? ???? ????????3E3 ={11}, P3 = 1/36
4.????? ????? ???????? ????????1???????? ???????? ????E4 ={11, 12, 13, 14, 15, 16}, P4 = 6/36
??????????? ??????? ????? ??????? ???? ????????? ????? ?????? ???? ???????
.?????????
6665646362616
5655545352515
4645444342414
3635343332313
2625242322212
1615141312111
654321-
:???????????? ????? ???????
???????? ????????? ????P (A B) = P (A) + P (B)
)??????? ?(???? ????????? ????? ??????? ?????? ????P (A B) = P (A) + P (B) - P (A B)
???? ?????? ???? ?????? ????? ????? ???????? ???????? ???? ???? ????? ?????? ??????? :??????4.
A = {1, 3, 5}, P (A) = 3/6
B = {5, 6}, P (B) = 2/6
AB = {5}, P (A B) = 1/6
P (A B) = 3/6 + 2/6 - 1/6 = 4/6
:??????
Male (M) Female
Smoking (S) 4 1 5
Not smoking 3 2 5
7 3 10
:?????? ????? ????? ???????? ???????? ????
P (M S) = P (M) + P (S) - P (M S) = 7/10 + 5/10 - 4/10 = 8/10

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( ???????? ??????????Conditional probability.)
( ??????? ????????? ??????? ??????? ??????? ??????? ???????? ???????? ????? ???????A)( ??????B.)(7)
( ??????? ????????A( ?????? )B)P (A / B) =
( ??????? ????????B( ?????? )A)P (B / A) =
= ????????? ??????????
???????? ?????????
???????? ???????
????????? ?????????:????????? ??????A, B????????? ???????.??????(8)
P (A / B) =
P (B / A) =
( :?????? ???? ????? ????A / B( = )B / A.?????? ?????? ???? ????? ????? ?)??
??????:????? ???????????? ???????)??????????? ????? ?????? ?(????????? ???? ???????? ??????:
1.?????????? ?????????(A)????????? ????4(B.)
P (A) = 6/36, P (B) = 3/36, P (A B) = 1/36
P (A or B) = P (A B) = P (A) + P (B) - P (A B) = 6/36 + 3/36 – 1/36 = 8/36
2.?????????? ?????????(A)??????????4(B.)
P (A) = 6/36, P (B) = 3/36, P (A B) = 8/36
P (A and B) = P (A B) = P (A) + P (B) - P (A B) = 6/36 + 3/36 - 8/36 = 1/36
3.?????????? ?????????(A)????????? ?????? ???? ??????4(B.)
P (A) = 6/36, P (B) = 3/36, P (A B) = 1/36
P (A/B) = = = 1/3
(7( ???????? )A and B???????? ????????? ???????? ???? )
(8???????? ??????? ?????? ?????? ???? ????? ???? ??????? ???? ???? ??????? ????? ?????? ???? ?????? ?????)????????? ?????? ??????? ????? ?????? ????? ??????? ????????? ?

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4.?????????4(B( ?????????? ????????? ?????? ?)A.)
P (B / A) = = = 1/6
5.?????????? ????????? ????
P (B / A) ≠ P (B) ? ????????? ????? ??????? ????? ?????????? ??????? ??????
( ???????????? ????? ???????Multiplication:)
?????????? ??????????????????? ????? ?????????
P (A B) = P (A) × P (B)P (A B) = P (B) × P (A / B)
P (A / B) = = P (A)P (A / B) =
:??????
?????????????
642??????
642??????
1284
:?????????
1.( ?????? ????????A( ?????? ?????? ????????? )B.)
P (A) = 4/12 = 1/3, P (B) = 6/12 = ? , P (A/ B) = 1/3 × ? = 1/6 (9)
P (A / B) = P (A B) / P (B) = = 1/3 = P (A) ????????? ?????????
????? ??????? :?????)???????????? ?(???? ???? ?????? )???????? ?(????????? ???? ?????? ????????????????????? ?????? ????
??????????????? ???????? ???? ??????????? ?????????? ??????? ???? ???????? ???????????????? ???????? ?????????? ???????
(9= ???????? ???? )2/12

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??????:????? ??????? ???????3??? ??????????? ??????????7?????? ????????????? ????? ????????????????? ?????? ??????? ?????????????????????????
??????? ???????????????????? ???????.(11)
1.??????? ????? ????? ?????????????
2.??????? ?????????? ?????????????
3.??????? ????????? ???????
:??????
????????
????????
???????
?????????
S
????????? ???? ?????????? ????????? ????? ???? ??????????
1st
2nd
P1 * P2 1st
2nd
R R RR 3/10 3/10 9/100 3/10 2/9 P(R) * P(R/R)
W RW 3/10 7/10 21/100 3/10 7/9 P(R) * P(W/R)
W R WR 7/10 3/10 21/100 7/10 3/9 P(W) * P(R/W)
W WW 7/10 7/10 49/100 7/10 6/9 P(W) * P(W/W)
???????? ???????? ?????
(11):????????????? ??????? ???????? ?????????? ????? ???????????????? ????? ????????? ?????? ?????????????????? ??????? ??????? ?????? ????????? ???? ????????? ?????? :??????? ?????? ?
.?????? ????????? ?????? ??????? ???? ????????? ??????????

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??????? ??????? ????????
1.????? ???????? ?????? ????? ????? ???????? ???????? ?????? ????? ?????? ???????2????3.
A ={2, 4, 6} P (A) = 3/6
B = {3, 6} P (B) = 2/6
P (A or B) = P (A) + P (B) - P (A B) = 3/6 + 2/6 - 1/6 = 4/6
2.:???????? ???????????? ?????? ???????????????? ????? ???????????? ???????
a..?????? ????????? ??????? ?????? ???? ????????
P (A) = 18 / 36
b.????( ?????????? ????????? ?????? ???? ??????A)???????????? ???6(B).
A = {11, 22, 33, 44, 55, 66} P (A) = 6/36
B = {15, 24, 33, 42, 51}
P (A or B) = 6/36 + 5/36 – 1/36 = 10/36
c.????????? ??????? ??????????6(A)?????????? ?????? ???? ???????? ????(B).
P (B / A) = 1/36 ÷ 5/36 = 1/5
3.???? ?????3???????????? ??????????7????????? ???????? ????????????????????????? ??????:??????
a.??????? ????????? ?????? ???? ????????(A)??????????? ???????????(B).
P (A and B) = 7/10 * 3/9 = 21/90
b..??????? ????? ???? ??????? ???? ????????
P (AA or BB) = 7/10 * 6/9 + 3/10 * 2/9 = 48/90
c.??????? ??????????????????.????????? ???????????? ??????
P (AB) = 7/10 * 3/10 = 21/100
P (AA or BB) = 7/10 * 6/10 + 3/10 * 2/10 = 48/100
4.:????? ?????? ???? ???????? ???? ???? ???????????????? ???????? ???????
S = {HH, HT, TH, TT}
a..???????? ???????? ???? ??????
P (A) = 2/4 = ?

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b..??????? ????? ????????? ??????
P (B) = 3/4
c..????????? ??????
P (C) = 2/4
d..?????????? ???????
P (D) = 2/4
5.:?????? ??????? ?????? ???????? ??????? ??????? ??????????? ???????? ????
a.??????????? ????????? ?????? ???? ????????(A)?????? ???????????(B).
P (A B) = 26/52 * 12/51 = 0.12
b.???? ????? ?????????? ???????? ???????????4(A)??????????? ?????? ???? ???????? ?????? ?(B).
P (A) = 12/52, P(B) = 26/52
P (B / A) = 6/52 ÷12/52 = ?
6.???????? ???? ???? ??????????? ???????????? ????? ?????????????? ??????? ?????? ????7????11.
A={16, 25, 34, 43, 52, 61} P (A) = 6/36
B = {46, 55, 64} P (B) = 3/36
P (A or B) = 6/36 + 3/36 = 9/36
7.:??????? ????? ??????? ?????? ???? ??????? ?????? ????
( ??????M)( ???????F)?????????
( ??????A)235
( ??????B)145
?????????3711
???? ????????? ??????? ?????????? .????????? ???????? ???? ???? ?????? ?????? ???? ???????? ???? ???? ???????????? ????? ????????
.??????? ?????? ???? ???????? ?????? ????????? ????????
P (M) = 3/10 P (A) = 5/10 P(A and M) = 2/10
P (M or A) = 3/10 + 5/10 - 2/10 = 6/10
P (F /B) = 4/10 ÷ 5/10 = 4/5

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8.???? ????? ????? ????? ???????? ???????? ?????? ??????? ?????? ???????3??????? ????(A)???????? ???? ?????? ????
(B)??????? ???? ?????? ????? ???????? ???? ?????? ???? ?????????? ?5(C).?????? ????? ???????? ???????? ??????
P (A) = 2/6
P (B) =1/2
P (C) = 2/6
Sheet (1)
??????? ????????:?????????????? ????? ???? ????? ????? ????? ??????????????.
1..????????? ????? ???????? ?????????? ????? ???????? ????? ????? ??????? ????? ????????? ??????????? ???? ??????????????? ?????????????
???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????
???????????????????????????????????????????????????
2.??????????(A)????? ????????? ????? ?????? ?????? ??????? ????? ????? ?????? ?????? ??????p(A)=0.5
??????P(A)=0???????? ?????????? ???????? ?????????? ???????? ?????? ???? ?????? ???????? ????? ????? ?
3.
??????? ????? = ????????? ???? ????? ??????????????????? ??????? ???? ????? ????????? ???? ????? ??????? ??????? ?????? ??????? ??????????
4.= ??????? ?????? ???? ?????? ?????? ???????? ??????????1.65????????????????????1.25.
????? ????????????????????? ???????≠1= ??????? ???????? ???? ????????? ?1.35
5.???????
?????????? ??????? ??????? ??????? ?
????????? ????????? ???????? ???? ???????????? ?:???? ???????? ????????A,B:
1.P(A B) = P(A) + P(B)
??????? ?????? ?????? ???????A???B?????????? ???????

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2.P(A|B) = P(A)
??????? ?????? ?????? ???????A???B????????????????

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???????? ????????:
??????????? ???????????:????? ????????? ????? ??????? ?????A={2,4,6 }, B={1,3,5}
????? ????A,B??? ?????????? ??????? ???????
:????? ??????? ??????? ?????P (A) + P (B) = S
????? = ????????? ????? ????????? ??????????
: ???????????? ????????
??????? ???????????????? ????? ???? ?????? ?????? ???????? ??????????1.6??????????? ???????? ???????? ??????? ?????? ??????1.1???????? ?????? ?
.?????? ??????? ?????? ??????
( ?????? ?????? ?????? ???? ???????A( ?????? ?????? ?????? ??????? ?)B):?????????? ?????? ?????????? ?
P (A) = 0.6, P (A B) = 0.1, P (A B’) = ?
P (A B’) = P (A) - P (A B) = 0.6 - 0.1 = 0.5
:???????? ????????
( ??????? ???? ??????????? ????? ???? ??????? ?????? ???? ???????? ????? ?????A)????1.35?????? ?????? ??????????? ????? ???? ???????????
???? ??????1.15( ??????? ???? ????? ??????? ???? ?????? ?????? ?????? ??????????? ????? ???? ???????????A???? )1.4
( :?????????1)?????? ?????? ?????? ?????( ?2( ????? ?????? )A( ??????? ?????? ?????? ?????? )3( ????? ??????? ???????? )A)
( ??????? ?????? ?????? ??????4?????????? ????? ???? )
( ?????????? ????????? ??????A?????? ??????? ?????? ???? )A( ?B?????? ????? ?????? ????????? ?????? )
P (A) = 0.35, P (B) = 0.15, P (A B) = 0.4
1. P (B’) = 0.85
2. P (A B’) = P (A) - P (A B) ????? ???????? ????? ?????????????????? ?????????
P (A B) = P(A) + P(B) - P (A B)
0.4 = 0.35 + 0.15 - P (A B), P (A B) = 0.1
P (A B’) = 0.35 - 0.1 = 0.25
3. | = 0.1/0.15 = 0.66

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4.:????????????? ?????? ????????? ????? ???? ???????? )????????? ?(????
P (A B) = P (A) × P (B)|
0.1 ≠ 0.65 × 0.15 = 0.09750.66 ≠ 0.65
???????? ????????:
:????????????? ????? ?????? ????? ???????? ????????? ?????? ????? ????????? ?????????????? ?????? ???????? ????????
????????? ??????
??????????????? ??????????(D)
??????????????????
511141???????????(A)
21113171???????
1519555???????
411235165?????????
( ????????? ???????? ????????????? ?????? ????????????? ??????? ????? ???????? ???? ????????????? ?????????A????? ????? ???????? ?????? )
( ???????????? ?????D???? ???????? ??????? ???? ????????? ??????? ????? ???????? ?????? ):????????? ???????
1..??????????? ????? ?????P (A) = 50/400
2..??????P (D) = 165/400
3..??????? ??????????? ????? ?????P (A D) = 40/400
4.. ?????? ?????? ?????? ??????????? ????? ?????| = 40/165
5.?????????? ????A, D????????????????? ???No, | P (A)
:???????? ????????????
1..????????? ???? ???????????? ?????? ???????? ?????? ?????????? ?????? ?????? ?????? ??????? ?????
???????(T???????? ?????? ?????? )
S = {TTT, TTH, THT. THH, HTT, HTH, HHT, HHH}
P (2T) = 3/8

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2.???????????? ?????? ?????????? ??????? ???????A , B??????? ????? ???????? ????????A????? ???????? ????? ???????? ??????????
???????(B)??????? ???? ???????? ?.?????A.??????????? ????????? ??????? ????? ??????? ????? ????? ??????
( ??????? ???????? ????? ???????? ???? ??????? ????A= )X( ???????? ???????? ????? ???????? ?????? ?B= )2X
P (A) = 1/3, P (B) = 2/3
S ={AA, AB, BA, BB}
P (W?1) = 3/4

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