:: Introduction to Probability :: by Jan Popio{\l}ek :: :: Received June 13, 1990 :: Copyright (c) 1990-2018 Association of Mizar Users :: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland). :: This code can be distributed under the GNU General Public Licence :: version 3.0 or later, or the Creative Commons Attribution-ShareAlike :: License version 3.0 or later, subject to the binding interpretation :: detailed in file COPYING.interpretation. :: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these :: licenses, or see http://www.gnu.org/licenses/gpl.html and :: http://creativecommons.org/licenses/by-sa/3.0/. environ vocabularies NUMBERS, XBOOLE_0, SUBSET_1, FINSEQ_1, TARSKI, FINSET_1, RELAT_1, CARD_1, ARYTM_3, XXREAL_0, REAL_1, ARYTM_1, RPR_1, BSPACE; notations TARSKI, XBOOLE_0, SUBSET_1, FUNCT_1, DOMAIN_1, ORDINAL1, CARD_1, NUMBERS, XCMPLX_0, REAL_1, XREAL_0, FINSEQ_1, FINSET_1, XXREAL_0; constructors XXREAL_0, REAL_1, NAT_1, MEMBERED, FINSEQ_1, DOMAIN_1, XREAL_0; registrations RELSET_1, FINSET_1, XXREAL_0, XREAL_0, CARD_1, ORDINAL1; requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM; begin reserve E for non empty set; reserve a for Element of E; reserve A, B for Subset of E; reserve Y for set; reserve p for FinSequence; theorem :: RPR_1:1 for e being non empty Subset of E holds e is Singleton of E iff for Y holds (Y c= e iff Y = {} or Y = e); registration let E; cluster -> finite for Singleton of E; end; reserve e, e1, e2 for Singleton of E; theorem :: RPR_1:2 e = A \/ B & A <> B implies A = {} & B = e or A = e & B = {}; theorem :: RPR_1:3 e = A \/ B implies A = e & B = e or A = e & B = {} or A = {} & B = e; theorem :: RPR_1:4 {a} is Singleton of E; theorem :: RPR_1:5 e1 c= e2 implies e1 = e2; theorem :: RPR_1:6 ex a st a in E & e = {a}; theorem :: RPR_1:7 ex e st e is Singleton of E; theorem :: RPR_1:8 ex p st p is FinSequence of E & rng p = e & len p = 1; definition let E be set; mode Event of E is Subset of E; end; theorem :: RPR_1:9 for E being non empty set, e being Singleton of E, A being Event of E holds e misses A or e /\ A = e; theorem :: RPR_1:10 for E being non empty set, A being Event of E st A <> {} ex e being Singleton of E st e c= A; theorem :: RPR_1:11 for E being non empty set, e being Singleton of E, A being Event of E st e c= A \/ A` holds e c= A or e c= A`; theorem :: RPR_1:12 e1 = e2 or e1 misses e2; theorem :: RPR_1:13 A /\ B misses A /\ B`; definition let E be finite set; let A be Event of E; func prob(A) -> Real equals :: RPR_1:def 1 card A / card E; end; theorem :: RPR_1:14 for E being finite non empty set, e being Singleton of E holds prob(e) = 1 / card E; theorem :: RPR_1:15 for E being finite non empty set holds prob([#] E) = 1; theorem :: RPR_1:16 for E being finite non empty set, A,B being Event of E st A misses B holds prob(A /\ B) = 0; theorem :: RPR_1:17 for E being finite non empty set, A being Event of E holds prob(A) <= 1; theorem :: RPR_1:18 for E being finite non empty set, A being Event of E holds 0 <= prob(A); theorem :: RPR_1:19 for E being finite non empty set, A,B being Event of E st A c= B holds prob(A) <= prob(B); theorem :: RPR_1:20 for E being finite non empty set, A,B being Event of E holds prob(A \/ B) = prob(A) + prob(B) - prob(A /\ B); theorem :: RPR_1:21 for E being finite non empty set, A,B being Event of E st A misses B holds prob(A \/ B) = prob(A) + prob(B); theorem :: RPR_1:22 for E being finite non empty set, A being Event of E holds prob( A) = 1 - prob(A`) & prob(A`) = 1 - prob(A); theorem :: RPR_1:23 for E being finite non empty set, A,B being Event of E holds prob(A \ B) = prob(A) - prob(A /\ B); theorem :: RPR_1:24 for E being finite non empty set, A,B being Event of E st B c= A holds prob(A \ B) = prob(A) - prob(B); theorem :: RPR_1:25 for E being finite non empty set, A,B being Event of E holds prob(A \/ B) <= prob(A) + prob(B); theorem :: RPR_1:26 for E being finite non empty set, A,B being Event of E holds prob(A) = prob(A /\ B) + prob(A /\ B`); theorem :: RPR_1:27 for E being finite non empty set, A,B being Event of E holds prob(A) = prob(A \/ B) - prob(B \ A); theorem :: RPR_1:28 for E being finite non empty set, A,B being Event of E holds prob(A) + prob(A` /\ B) = prob(B) + prob(B` /\ A); theorem :: RPR_1:29 for E being finite non empty set, A,B,C being Event of E holds prob(A \/ B \/ C) = ( prob(A) + prob(B) + prob(C) ) - ( prob(A /\ B) + prob(A /\ C) + prob(B /\ C) ) + prob(A /\ B /\ C); theorem :: RPR_1:30 for E being finite non empty set, A,B,C being Event of E st A misses B & A misses C & B misses C holds prob(A \/ B \/ C) = prob(A) + prob(B) + prob(C) ; theorem :: RPR_1:31 for E being finite non empty set, A,B being Event of E holds prob(A) - prob(B) <= prob(A \ B); definition let E be finite set; let B,A be Event of E; func prob(A, B) -> Real equals :: RPR_1:def 2 prob(A /\ B) / prob(B); end; theorem :: RPR_1:32 for E being finite non empty set, A being Event of E holds prob(A, [#]E ) = prob(A); theorem :: RPR_1:33 for E being finite non empty set holds prob([#] E, [#] E) = 1; theorem :: RPR_1:34 for E being finite non empty set, A,B being Event of E st 0 < prob(B) holds prob(A, B) <= 1; theorem :: RPR_1:35 for E being finite non empty set, A,B being Event of E st 0 < prob(B) holds 0 <= prob(A, B); theorem :: RPR_1:36 for E being finite non empty set, A,B being Event of E st 0 < prob(B) holds prob(A, B) = 1 - prob(B \ A) / prob(B); theorem :: RPR_1:37 for E being finite non empty set, A,B being Event of E st 0 < prob(B) & A c= B holds prob(A, B) = prob(A) / prob(B); theorem :: RPR_1:38 for E being finite non empty set, A,B being Event of E st A misses B holds prob(A, B) = 0; theorem :: RPR_1:39 for E being finite non empty set, A,B being Event of E st 0 < prob(A) & 0 < prob(B) holds prob(A) * prob(B, A) = prob(B) * prob(A, B); theorem :: RPR_1:40 for E being finite non empty set, A,B being Event of E st 0 < prob B holds prob(A, B) = 1 - prob(A`, B) & prob(A`, B) = 1 - prob(A, B); theorem :: RPR_1:41 for E being finite non empty set, A,B being Event of E st 0 < prob(B) & B c= A holds prob(A, B) = 1; theorem :: RPR_1:42 for E being finite non empty set, B being Event of E st 0 < prob(B) holds prob([#] E, B) = 1; theorem :: RPR_1:43 for E being finite non empty set, A being Event of E holds prob(A`, A) = 0; theorem :: RPR_1:44 for E being finite non empty set, A being Event of E holds prob(A, A`) = 0; theorem :: RPR_1:45 for E being finite non empty set, A,B being Event of E st 0 < prob(B) & A misses B holds prob(A`, B) = 1; theorem :: RPR_1:46 for E being finite non empty set, A,B being Event of E st 0 < prob(A) & prob(B) < 1 & A misses B holds prob(A, B`) = prob(A) / (1 - prob(B)); theorem :: RPR_1:47 for E being finite non empty set, A,B being Event of E st 0 < prob(A) & prob(B) < 1 & A misses B holds prob(A`, B`) = 1 - prob(A) / (1 - prob(B)); theorem :: RPR_1:48 for E being finite non empty set, A,B,C being Event of E st 0 < prob(B /\ C) & 0 < prob(C) holds prob(A /\ B /\ C) = prob(A, B /\ C) * prob(B, C) * prob(C); theorem :: RPR_1:49 for E being finite non empty set, A,B being Event of E st 0 < prob(B) & prob(B) < 1 holds prob(A) = prob(A, B) * prob(B) + prob(A, B`) * prob (B`); theorem :: RPR_1:50 for E being finite non empty set, A,B1,B2 being Event of E st 0 < prob(B1) & 0 < prob(B2) & B1 \/ B2 = E & B1 misses B2 holds prob(A) = prob(A, B1) * prob(B1) + prob(A, B2) * prob(B2); theorem :: RPR_1:51 for E being finite non empty set, A,B1,B2,B3 being Event of E st 0 < prob(B1) & 0 < prob(B2) & 0 < prob(B3) & B1 \/ B2 \/ B3 = E & B1 misses B2 & B1 misses B3 & B2 misses B3 holds prob(A) = ( prob(A, B1) * prob(B1) + prob(A , B2) * prob(B2) ) + prob(A, B3) * prob(B3); theorem :: RPR_1:52 for E being finite non empty set, A,B1,B2 being Event of E st 0 < prob (B1) & 0 < prob(B2) & B1 \/ B2 = E & B1 misses B2 holds prob(B1, A) = ( prob(A, B1) * prob(B1) ) / ( prob(A, B1) * prob(B1) + prob(A, B2) * prob(B2) ); theorem :: RPR_1:53 for E being finite non empty set, A,B1,B2,B3 being Event of E st 0 < prob(B1) & 0 < prob(B2) & 0 < prob(B3) & B1 \/ B2 \/ B3 = E & B1 misses B2 & B1 misses B3 & B2 misses B3 holds prob(B1, A) = ( prob(A, B1) * prob(B1) ) / ( ( prob(A, B1) * prob(B1) + prob(A, B2) * prob(B2) ) + prob(A, B3) * prob(B3) ); definition let E be finite set; let A, B be Event of E; pred A, B are_independent means :: RPR_1:def 3 prob(A /\ B) = prob(A) * prob(B); symmetry; end; theorem :: RPR_1:54 for E being finite non empty set, A,B being Event of E st 0 < prob(B) & A, B are_independent holds prob(A, B) = prob(A); theorem :: RPR_1:55 for E being finite non empty set, A,B being Event of E st prob(B) = 0 holds A, B are_independent; theorem :: RPR_1:56 for E being finite non empty set, A,B being Event of E st A, B are_independent holds A`, B are_independent; theorem :: RPR_1:57 for E being finite non empty set, A,B being Event of E st A misses B & A, B are_independent holds prob(A) = 0 or prob(B) = 0;