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As 506 f1 ??????????

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Ahmed El Kordy
Ahmed El Kordy

probability statistics lecture

1 of 14
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Page 1 of 14 .??????? ???? ??????? ?????????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??????? ????? ????? ).?????????? ???????? ??????? ???????? ??????? ?????? ????? ??????? ????? ?????????? ????????????? ??????????? ????? ????? ???????? ?????????
Page 2 of 14 ???????? ???????? ???????? ?????????? ?????????????:???????? 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
Page 3 of 14 :????????? ???????? ????????? ????? ??????? ?????? ??????????????? ??????? 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%???? ?????????? ???????? ???????????? ????????????? ????? ?.????????? ???? ????????????
Page 4 of 14 :????????????????? ????? ?????? ??????? 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
Page 5 of 14 ( ???????? ??????????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 C 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???????? ??????? ?????? ?????? ???? ????? ???? ??????? ???? ???? ??????? ????? ?????? ???? ?????? ?????)????????? ?????? ??????? ????? ?????? ????? ??????? ????????? ?
Page 6 of 14 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
Page 7 of 14 ??????:????? ??????? ???????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):????????????? ??????? ???????? ?????????? ????? ???????????????? ????? ????????? ?????? ?????????????????? ??????? ??????? ?????? ????????? ???? ????????? ?????? :??????? ?????? ? .?????? ????????? ?????? ??????? ???? ????????? ??????????
Page 8 of 14 ??????? ??????? ???????? 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 C 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 = ?
Page 9 of 14 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
Page 10 of 14 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?????????? ???????
Page 11 of 14 2.P(A|B) = P(A) ??????? ?????? ?????? ???????A???B????????????????
Page 12 of 14 ???????? ????????: ??????????? ???????????:????? ????????? ????? ??????? ?????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
Page 13 of 14 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
Page 14 of 14 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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As 506 f1 ??????????

  • 1. Page 1 of 14 .??????? ???? ??????? ?????????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??????? ????? ????? ).?????????? ???????? ??????? ???????? ??????? ?????? ????? ??????? ????? ?????????? ????????????? ??????????? ????? ????? ???????? ?????????
  • 2. Page 2 of 14 ???????? ???????? ???????? ?????????? ?????????????:???????? 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
  • 3. Page 3 of 14 :????????? ???????? ????????? ????? ??????? ?????? ??????????????? ??????? 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%???? ?????????? ???????? ???????????? ????????????? ????? ?.????????? ???? ????????????
  • 4. Page 4 of 14 :????????????????? ????? ?????? ??????? 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
  • 5. Page 5 of 14 ( ???????? ??????????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 C 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???????? ??????? ?????? ?????? ???? ????? ???? ??????? ???? ???? ??????? ????? ?????? ???? ?????? ?????)????????? ?????? ??????? ????? ?????? ????? ??????? ????????? ?
  • 6. Page 6 of 14 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
  • 7. Page 7 of 14 ??????:????? ??????? ???????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):????????????? ??????? ???????? ?????????? ????? ???????????????? ????? ????????? ?????? ?????????????????? ??????? ??????? ?????? ????????? ???? ????????? ?????? :??????? ?????? ? .?????? ????????? ?????? ??????? ???? ????????? ??????????
  • 8. Page 8 of 14 ??????? ??????? ???????? 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 C 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 = ?
  • 9. Page 9 of 14 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
  • 10. Page 10 of 14 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?????????? ???????
  • 11. Page 11 of 14 2.P(A|B) = P(A) ??????? ?????? ?????? ???????A???B????????????????
  • 12. Page 12 of 14 ???????? ????????: ??????????? ???????????:????? ????????? ????? ??????? ?????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
  • 13. Page 13 of 14 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
  • 14. Page 14 of 14 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