:: Introduction to Probability
:: by Jan Popio{\l}ek
::
:: Received June 13, 1990
:: Copyright (c) 1990-2026 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;
definitions XBOOLE_0;
equalities XBOOLE_0, SUBSET_1;
expansions XBOOLE_0;
theorems TARSKI, SUBSET_1, ZFMISC_1, FINSEQ_1, CARD_1, CARD_2, XBOOLE_0,
XBOOLE_1, XCMPLX_0, XCMPLX_1, XREAL_1, NAT_1;
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 Th1:
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)
proof
let e be non empty Subset of E;
thus e is Singleton of E implies for Y holds (Y c= e iff Y = {} or Y = e)
proof
assume
A1: e is Singleton of E;
let Y;
ex x being object st e = {x} by A1,ZFMISC_1:131;
hence thesis by ZFMISC_1:33;
end;
assume
A2: for Y holds Y c= e iff Y = {} or Y = e;
consider x being object such that
A3: x in e by XBOOLE_0:def 1;
{x} c= e by A3,ZFMISC_1:31;
hence thesis by A2;
end;
registration
let E;
cluster -> finite for Singleton of E;
coherence;
end;
reserve e, e1, e2 for Singleton of E;
theorem
e = A \/ B & A <> B implies A = {} & B = e or A = e & B = {}
proof
assume that
A1: e = A \/ B and
A2: A <> B;
A c= e by A1,XBOOLE_1:7;
then
A3: A = {} or A = e by Th1;
B c= e by A1,XBOOLE_1:7;
hence thesis by A2,A3,Th1;
end;
theorem
e = A \/ B implies A = e & B = e or A = e & B = {} or A = {} & B = e
proof
assume
A1: e = A \/ B;
then A c= e & B c= e by XBOOLE_1:7;
then
A = {} & B = e or A = e & B = {} or A = e & B = e or A = {} & B = {} by Th1;
hence thesis by A1;
end;
theorem
{a} is Singleton of E;
theorem
e1 c= e2 implies e1 = e2 by Th1;
theorem Th6:
ex a st a in E & e = {a}
proof
set x = the Element of e;
{x} = e by Th1;
hence thesis;
end;
theorem
ex e st e is Singleton of E
proof
take {the Element of E};
thus thesis;
end;
theorem
ex p st p is FinSequence of E & rng p = e & len p = 1
proof
consider a such that
a in E and
A1: e = {a} by Th6;
rng <*a*> = {a} & len <*a*> = 1 by FINSEQ_1:39;
hence thesis by A1;
end;
definition
let E be set;
mode Event of E is Subset of E;
end;
theorem
for E being non empty set, e being Singleton of E, A being Event of E
holds e misses A or e /\ A = e
proof
let E be non empty set, e be Singleton of E, A be Event of E;
e /\ E = e & A \/ A` = [#] E by SUBSET_1:10,XBOOLE_1:28;
then e = e /\ A \/ e /\ A` by XBOOLE_1:23;
then e /\ A c= e by XBOOLE_1:7;
then e /\ A = {} or e /\ A = e by Th1;
hence thesis;
end;
theorem
for E being non empty set, A being Event of E st A <> {} ex e being
Singleton of E st e c= A
proof
let E be non empty set, A be Event of E;
set x = the Element of A;
assume
A1: A <> {};
then reconsider x as Element of E by TARSKI:def 3;
{x} c= A by A1,ZFMISC_1:31;
hence thesis;
end;
theorem
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`
proof
let E be non empty set, e be Singleton of E, A be Event of E;
ex a being Element of E st a in E & e = {a} by Th6;
then consider a being Element of E such that
A1: e = {a};
assume e c= A \/ A`;
then a in A \/ A` by A1,ZFMISC_1:31;
then a in A or a in A` by XBOOLE_0:def 3;
hence thesis by A1,ZFMISC_1:31;
end;
theorem
e1 = e2 or e1 misses e2
proof
e1 /\ e2 c= e1 by XBOOLE_1:17;
then e1 /\ e2 = {} or e1 /\ e2 = e1 by Th1;
then e1 c= e2 or e1 /\ e2 = {} by XBOOLE_1:17;
hence thesis by Th1;
end;
theorem Th13:
A /\ B misses A /\ B`
proof
A /\ B misses A \ B by XBOOLE_1:89;
hence thesis by SUBSET_1:13;
end;
Lm1: for E being finite non empty set holds 0 < card E
proof
let E be finite non empty set;
card {the Element of E} <= card E by NAT_1:43;
hence thesis by CARD_1:30;
end;
definition
let E be finite set;
let A be Event of E;
func prob(A) -> Real equals
card A / card E;
coherence;
end;
theorem
for E being finite non empty set, e being Singleton of E holds prob(e) = 1
/ card E by CARD_1:def 7;
theorem
for E being finite non empty set holds prob([#] E) = 1 by XCMPLX_1:60;
theorem Th16:
for E being finite non empty set, A,B being Event of E st A
misses B holds prob(A /\ B) = 0
by CARD_1:27;
theorem
for E being finite non empty set, A being Event of E holds prob(A) <= 1
proof
let E be finite non empty set, A be Event of E;
0 < card E by Lm1;
then card A * (card E)" <= card E * (card E)" by NAT_1:43,XREAL_1:64;
then card A / card E <= card E * (card E)" by XCMPLX_0:def 9;
then prob([#] E) = card E / card E & prob(A) <= card E / card E by
XCMPLX_0:def 9;
hence thesis by XCMPLX_1:60;
end;
theorem Th18:
for E being finite non empty set, A being Event of E holds 0 <= prob(A)
proof
let E be finite non empty set, A be Event of E;
0 < card E & 0 <= card A by Lm1,CARD_1:27;
hence thesis;
end;
theorem Th19:
for E being finite non empty set, A,B being Event of E st A c= B
holds prob(A) <= prob(B)
proof
let E be finite non empty set, A,B be Event of E;
assume
A1: A c= B;
0 < card E by Lm1;
then card A * (card E)" <= card B * (card E)" by A1,NAT_1:43,XREAL_1:64;
then card A / card E <= card B * (card E)" by XCMPLX_0:def 9;
hence thesis by XCMPLX_0:def 9;
end;
theorem Th20:
for E being finite non empty set, A,B being Event of E holds
prob(A \/ B) = prob(A) + prob(B) - prob(A /\ B)
proof
let E be finite non empty set, A,B be Event of E;
set q = ( card E )";
set p = card E;
card (( A \/ B ) qua Event of E) = card A + card B - card ( A /\ B ) by
CARD_2:45;
then
card ( A \/ B ) * q = card A * q + ( card B * q - card ( A /\ B ) * q );
then card ( A \/ B ) / p = card A * q + card B * q - card ( A /\ B ) * q by
XCMPLX_0:def 9;
then card ( A \/ B ) / p = card A / p + card B * q - card ( A /\ B ) * q by
XCMPLX_0:def 9;
then card ( A \/ B ) / p = card A / p + card B / p - card ( A /\ B ) * q by
XCMPLX_0:def 9;
hence thesis by XCMPLX_0:def 9;
end;
theorem Th21:
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)
proof
let E be finite non empty set, A,B be Event of E;
assume A misses B;
then prob(A /\ B) = 0 by Th16;
then prob(A \/ B) = prob(A) + prob(B) - 0 by Th20;
hence thesis;
end;
theorem Th22:
for E being finite non empty set, A being Event of E holds prob(
A) = 1 - prob(A`) & prob(A`) = 1 - prob(A)
proof
let E be finite non empty set, A be Event of E;
A misses A` by SUBSET_1:24;
then prob(A \/ A`) = prob(A) + prob(A`) by Th21;
then prob( [#] E ) = prob(A) + prob(A`) by SUBSET_1:10;
then 1 = prob(A) + prob(A`) by XCMPLX_1:60;
hence thesis;
end;
theorem Th23:
for E being finite non empty set, A,B being Event of E holds
prob(A \ B) = prob(A) - prob(A /\ B)
proof
let E be finite non empty set, A,B be Event of E;
prob(A) = prob((A \ B) \/ (A /\ B)) by XBOOLE_1:51;
then prob(A) = prob(A \ B) + prob(A /\ B) by Th21,XBOOLE_1:89;
hence thesis;
end;
theorem Th24:
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)
proof
let E be finite non empty set, A,B be Event of E;
assume B c= A;
then prob(A /\ B) = prob(B) by XBOOLE_1:28;
hence thesis by Th23;
end;
theorem
for E being finite non empty set, A,B being Event of E holds prob(A \/
B) <= prob(A) + prob(B)
proof
let E be finite non empty set, A,B be Event of E;
prob(A \/ B) = prob(A) + prob(B) - prob(A /\ B) by Th20;
hence thesis by Th18,XREAL_1:43;
end;
theorem Th26:
for E being finite non empty set, A,B being Event of E holds
prob(A) = prob(A /\ B) + prob(A /\ B`)
proof
let E be finite non empty set, A,B be Event of E;
A = A /\ ( A \/ [#] E ) by XBOOLE_1:21;
then A = A /\ [#] E by SUBSET_1:11;
then
A1: A = A /\ ( B \/ B`) by SUBSET_1:10;
prob((A /\ B) \/ (A /\ B`)) = prob(A /\ B) + prob(A /\ B`) by Th13,Th21;
hence thesis by A1,XBOOLE_1:23;
end;
theorem
for E being finite non empty set, A,B being Event of E holds prob(A) =
prob(A \/ B) - prob(B \ A)
proof
let E be finite non empty set, A,B be Event of E;
prob(A \/ (B \ A)) = prob(A \/ B) by XBOOLE_1:39;
then prob(A \/ B) = prob(A) + prob(B \ A) by Th21,XBOOLE_1:79;
hence thesis;
end;
theorem
for E being finite non empty set, A,B being Event of E holds prob(A) +
prob(A` /\ B) = prob(B) + prob(B` /\ A)
proof
let E be finite non empty set, A,B be Event of E;
prob(A) = prob(A /\ B) + prob(A /\ B`) & prob(B) = prob(A /\ B) + prob(B
/\ A`) by Th26;
hence thesis;
end;
theorem Th29:
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)
proof
let E be finite non empty set, A,B,C be Event of E;
prob(A \/ B \/ C) = prob(A \/ B) + prob(C) - prob((A \/ B) /\ C) by Th20
.= ( ( prob(A) + prob(B) ) - prob(A /\ B) ) + prob(C) - prob((A \/ B) /\
C) by Th20
.= ( prob(A) + prob(B) + prob(C) ) + -prob(A /\ B) - prob((A /\ C) \/ (B
/\ C)) by XBOOLE_1:23
.= ( prob(A) + prob(B) + prob(C) ) + -prob(A /\ B) - ( prob(A /\ C) +
prob(B /\ C) - prob((A /\ C) /\ (B /\ C)) ) by Th20
.= ( prob(A) + prob(B) + prob(C) ) + -prob(A /\ B) - ( prob(A /\ C) +
prob(B /\ C) - prob(A /\ ( C /\ (C /\ B)) )) by XBOOLE_1:16
.= ( prob(A) + prob(B) + prob(C) ) + -prob(A /\ B) - ( prob(A /\ C) +
prob(B /\ C) - prob(A /\ (( C /\ C ) /\ B) )) by XBOOLE_1:16
.= (( prob(A) + prob(B) + prob(C) ) + -prob(A /\ B)) - ( prob(A /\ C) +
prob(B /\ C) - prob(A /\ B /\ C) ) by XBOOLE_1:16
.= ( prob(A) + prob(B) + prob(C) ) + -( prob(A /\ B) + prob(A /\ C) +
prob(B /\ C) ) + prob(A /\ B /\ C);
hence thesis;
end;
theorem
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)
proof
let E be finite non empty set, A,B,C be Event of E;
assume that
A1: A misses B and
A2: A misses C and
A3: B misses C;
A4: prob(A /\ (B /\ C)) = 0 by A1,Th16,XBOOLE_1:74;
prob(A \/ B \/ C) = ( prob(A) + prob(B) + prob(C) ) - ( prob(A /\ B) +
prob(A /\ C) + prob(B /\ C) ) + prob(A /\ B /\ C) by Th29
.= ( prob(A) + prob(B) + prob(C) ) - ( prob(A /\ B) + prob(A /\ C) +
prob(B /\ C) ) + 0 by A4,XBOOLE_1:16
.= ( prob(A) + prob(B) + prob(C) ) - ( prob(A /\ B) + prob(A /\ C) + 0 )
by A3,Th16
.= ( prob(A) + prob(B) + prob(C) ) - ( prob(A /\ B) + 0 ) by A2,Th16
.= ( prob(A) + prob(B) + prob(C) ) - 0 by A1,Th16;
hence thesis;
end;
theorem
for E being finite non empty set, A,B being Event of E holds prob(A) -
prob(B) <= prob(A \ B)
proof
let E be finite non empty set, A,B be Event of E;
prob(A /\ B) <= prob(B) by Th19,XBOOLE_1:17;
then prob(A) - prob(B) <= prob(A) - prob(A /\ B) by XREAL_1:13;
hence thesis by Th23;
end;
definition
let E be finite set;
let B,A be Event of E;
func prob(A, B) -> Real equals
prob(A /\ B) / prob(B);
coherence;
end;
theorem
for E being finite non empty set, A being Event of E holds
prob(A, [#]E ) = prob(A)
proof
let E be finite non empty set, A be Event of E;
prob([#] E) = 1 by XCMPLX_1:60;
hence thesis by XBOOLE_1:28;
end;
theorem
for E being finite non empty set holds prob([#] E, [#] E) = 1
proof
let E be finite non empty set;
prob([#] E) = 1 by XCMPLX_1:60;
hence thesis;
end;
theorem
for E being finite non empty set, A,B being Event of E st 0 < prob(B)
holds prob(A, B) <= 1
proof
let E be finite non empty set, A,B be Event of E;
assume
A1: 0 < prob(B);
A /\ B c= B by XBOOLE_1:17;
then prob(A /\ B) * (prob(B))" <= prob(B) * (prob(B))" by A1,Th19,XREAL_1:64;
then prob(A /\ B) * (prob(B))" <= 1 by A1,XCMPLX_0:def 7;
hence thesis by XCMPLX_0:def 9;
end;
theorem
for E being finite non empty set, A,B being Event of E st 0 < prob(B)
holds 0 <= prob(A, B)
proof
let E be finite non empty set, A,B be Event of E;
assume
A1: 0 < prob(B);
0 <= prob(A /\ B) by Th18;
hence thesis by A1;
end;
theorem Th36:
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)
proof
let E be finite non empty set, A,B be Event of E;
prob(B \ A) + prob(A /\ B) = ( prob(B) - prob(A /\ B) ) + prob(A /\ B)
by Th23;
then prob(A, B) = ( prob(B) - prob(B \ A) ) / prob(B);
then
A1: prob(A, B) = prob(B) / prob(B) - prob(B \ A) / prob(B) by XCMPLX_1:120;
assume 0 < prob(B);
hence thesis by A1,XCMPLX_1:60;
end;
theorem
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)
proof
let E be finite non empty set, A,B be Event of E;
assume that
A1: 0 < prob(B) and
A2: A c= B;
prob(A, B) = 1 - prob(B \ A) / prob(B) by A1,Th36;
then prob(A, B) = 1 - ( prob(B) - prob(A) ) / prob(B) by A2,Th24;
then prob(A, B) = 1 - ( prob(B) / prob(B) - prob(A) / prob(B) ) by
XCMPLX_1:120;
then prob(A, B) = 1 - ( 1 - prob(A) / prob(B) ) by A1,XCMPLX_1:60;
hence thesis;
end;
theorem Th38:
for E being finite non empty set, A,B being Event of E st A
misses B holds prob(A, B) = 0
proof
let E be finite non empty set, A,B be Event of E;
assume A misses B;
then prob(A, B) = 0 / prob(B) by Th16
.= 0 * (prob(B))";
hence thesis;
end;
theorem Th39:
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)
proof
let E be finite non empty set, A,B be Event of E;
assume that
A1: 0 < prob(A) and
A2: 0 < prob(B);
prob(A) * prob(B, A) = prob(A /\ B) by A1,XCMPLX_1:87;
hence thesis by A2,XCMPLX_1:87;
end;
theorem Th40:
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)
proof
let E be finite non empty set, A,B be Event of E;
assume
A1: 0 < prob(B);
(A \/ A`) /\ B = [#] E /\ B & [#] E /\ B = B by SUBSET_1:10,XBOOLE_1:28;
then (A /\ B) \/ (A` /\ B) = B by XBOOLE_1:23;
then prob(A /\ B) + prob(A` /\ B) = prob(B) by Th13,Th21;
then prob(A, B) * prob(B) + prob(A` /\ B) = prob(B) by A1,XCMPLX_1:87;
then prob(A, B) * prob(B) + prob(A`, B) * prob(B) = prob(B) by A1,XCMPLX_1:87
;
then ( prob(A, B) + prob(A` , B) ) * prob(B) * (prob(B))" = 1 by A1,
XCMPLX_0:def 7;
then ( prob(A, B) + prob(A`, B) ) * ( prob(B) * (prob(B))" ) = 1;
then ( prob(A, B) + prob(A`, B) ) * 1 = 1 by A1,XCMPLX_0:def 7;
hence thesis;
end;
theorem Th41:
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
proof
let E be finite non empty set, A,B be Event of E;
assume that
A1: 0 < prob(B) and
A2: B c= A;
prob(A /\ B) = prob(B) by A2,XBOOLE_1:28;
hence thesis by A1,XCMPLX_1:60;
end;
theorem
for E being finite non empty set, B being Event of E st 0 < prob(B)
holds prob([#] E, B) = 1 by Th41;
theorem
for E being finite non empty set, A being Event of E holds prob(A`, A) = 0
proof
let E be finite non empty set, A be Event of E;
A` misses A by SUBSET_1:24;
then prob(A` /\ A) = 0 by Th16;
hence thesis;
end;
theorem
for E being finite non empty set, A being Event of E holds prob(A, A`) = 0
proof
let E be finite non empty set, A be Event of E;
A misses A` by SUBSET_1:24;
then prob(A /\ A`) = 0 by Th16;
hence thesis;
end;
theorem Th45:
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
proof
let E be finite non empty set, A,B be Event of E;
assume that
A1: 0 < prob(B) and
A2: A misses B;
prob(A, B) = 0 by A2,Th38;
then 1 - prob(A`, B) = 0 by A1,Th40;
hence thesis;
end;
theorem Th46:
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))
proof
let E be finite non empty set, A,B be Event of E;
assume that
A1: 0 < prob(A) and
A2: prob(B) < 1 and
A3: A misses B;
prob(B) - 1 < 1 - 1 by A2,XREAL_1:9;
then 0 < - ( - ( 1 - prob(B) ) );
then
A4: 0 < prob(B`) by Th22;
then prob(A) * prob(B`, A) = prob(B`) * prob(A, B`) by A1,Th39;
then prob(A) * 1 = prob(B`) * prob(A, B`) by A1,A3,Th45;
then prob(A) * (prob(B`))" = prob(A, B`) * ( prob(B`) * (prob(B`))" );
then
A5: prob(A) * (prob(B`))" = prob(A, B`) * 1 by A4,XCMPLX_0:def 7;
prob(B`) = 1 - prob(B) by Th22;
hence thesis by A5,XCMPLX_0:def 9;
end;
theorem
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))
proof
let E be finite non empty set, A,B be Event of E;
assume that
A1: 0 < prob(A) and
A2: prob(B) < 1 and
A3: A misses B;
A4: prob(B`) = 1 - prob(B) by Th22;
prob(B) -1 < 1 - 1 by A2,XREAL_1:9;
then 0 < - ( - ( 1 - prob(B) ) );
then prob(A` , B`) = 1 - prob(A, B`) by A4,Th40;
hence thesis by A1,A2,A3,Th46;
end;
theorem
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)
proof
let E be finite non empty set, A,B,C be Event of E;
assume that
A1: 0 < prob(B /\ C) and
A2: 0 < prob(C);
A3: prob(B /\ C) = prob(B, C) * prob(C) by A2,XCMPLX_1:87;
prob(A /\ B /\ C) = prob(A /\ (B /\ C)) by XBOOLE_1:16;
then prob(A /\ B /\ C) = prob(A, B /\ C) * prob(B /\ C) by A1,XCMPLX_1:87;
hence thesis by A3;
end;
theorem Th49:
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`)
proof
let E be finite non empty set, A,B be Event of E;
assume that
A1: 0 < prob(B) and
A2: prob(B) < 1;
prob(B) -1 < 1 - 1 by A2,XREAL_1:9;
then 0 < - ( - ( 1 - prob(B) ) );
then
A3: 0 < prob(B`) by Th22;
prob(A) = prob(A /\ B) + prob(A /\ B`) by Th26;
then prob(A) = prob(A, B) * prob(B) + prob(A /\ B`) by A1,XCMPLX_1:87;
hence thesis by A3,XCMPLX_1:87;
end;
theorem Th50:
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)
proof
let E be finite non empty set, A,B1,B2 be Event of E;
assume that
A1: 0 < prob(B1) and
A2: 0 < prob(B2) and
A3: B1 \/ B2 = E and
A4: B1 misses B2;
A5: B2 \ B1 = E \ B1 by A3,XBOOLE_1:40;
then 0 < prob((B1)`) by A2,A4,XBOOLE_1:83;
then 0 < 1 - prob(B1) by Th22;
then
A6: 1 - ( 1 - prob(B1) ) < 1 by XREAL_1:44;
B2 = B1` by A4,A5,XBOOLE_1:83;
hence thesis by A1,A6,Th49;
end;
theorem Th51:
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)
proof
let E be finite non empty set, A,B1,B2,B3 be Event of E;
assume that
A1: 0 < prob(B1) and
A2: 0 < prob(B2) and
A3: 0 < prob(B3) and
A4: B1 \/ B2 \/ B3 = E and
A5: B1 /\ B2 = {} and
A6: B1 /\ B3 = {} and
A7: B2 /\ B3 = {};
(B1 /\ B3) \/ (B2 /\ B3) = B2 /\ B3 by A6;
then (B1 \/ B2) /\ B3 = {} by A7,XBOOLE_1:23;
then
A8: (B1 \/ B2) misses B3;
(B1 \/ B2 \/ B3) /\ A = A by A4,XBOOLE_1:28;
then ((B1 \/ B2) /\ A) \/ (B3 /\ A) = A by XBOOLE_1:23;
then prob(A) = prob((B1 \/ B2) /\ A) + prob(B3 /\ A) by A8,Th21,XBOOLE_1:76;
then
A9: prob(A) = prob((B1 /\ A) \/ (B2 /\ A)) + prob(B3 /\ A) by XBOOLE_1:23;
B1 misses B2 by A5;
then prob(A) = prob(A /\ B1) + prob(A /\ B2) + prob(A /\ B3) by A9,Th21,
XBOOLE_1:76;
then
prob(A) = prob(A, B1) * prob(B1) + prob(A /\ B2) + prob(A /\ B3) by A1,
XCMPLX_1:87;
then prob(A) = prob(A, B1) * prob(B1) + prob(A, B2) * prob(B2) + prob(A /\
B3) by A2,XCMPLX_1:87;
hence thesis by A3,XCMPLX_1:87;
end;
theorem
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) )
proof
let E be finite non empty set, A,B1,B2 be Event of E;
assume that
A1: 0 < prob(B1) and
A2: 0 < prob(B2) & B1 \/ B2 = E & B1 misses B2;
prob(A) = prob(A, B1) * prob(B1) + prob(A, B2) * prob(B2) by A1,A2,Th50;
hence thesis by A1,XCMPLX_1:87;
end;
theorem
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) )
proof
let E be finite non empty set, A,B1,B2,B3 be Event of E;
assume that
A1: 0 < prob(B1) and
A2: 0 < prob(B2) & 0 < prob(B3) & B1 \/ B2 \/ B3 = E & B1 misses B2 & B1
misses B3 & B2 misses B3;
prob(A) = ( prob(A, B1) * prob(B1) + prob(A, B2) * prob(B2) ) + prob(A,
B3) * prob(B3) by A1,A2,Th51;
hence thesis by A1,XCMPLX_1:87;
end;
definition
let E be finite set;
let A, B be Event of E;
pred A, B are_independent means
prob(A /\ B) = prob(A) * prob(B);
symmetry;
end;
theorem
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)
proof
let E be finite non empty set, A,B be Event of E;
assume that
A1: 0 < prob(B) and
A2: A, B are_independent;
prob(A /\ B) = prob(A) * prob(B) by A2;
then prob(A, B) = prob(A) * ( prob(B) / prob(B) ) by XCMPLX_1:74;
then prob(A, B) = prob(A) * 1 by A1,XCMPLX_1:60;
hence thesis;
end;
theorem
for E being finite non empty set, A,B being Event of E st prob(B) = 0
holds A, B are_independent
proof
let E be finite non empty set, A,B be Event of E;
assume
A1: prob(B) = 0;
then prob(A /\ B) <= 0 by Th19,XBOOLE_1:17;
then prob(A /\ B) = 0 by Th18;
hence thesis by A1;
end;
theorem
for E being finite non empty set, A,B being Event of E st A, B
are_independent holds A`, B are_independent
proof
let E be finite non empty set, A,B be Event of E;
prob(A` /\ B) = prob(B \ A) by SUBSET_1:13;
then
A1: prob(A` /\ B) = prob(B) - prob(A /\ B) by Th23;
assume A, B are_independent;
then prob(A` /\ B) = 1 * prob(B) - prob(A) * prob(B) by A1;
then prob(A` /\ B) = ( 1 - prob(A) ) * prob(B);
then prob(A` /\ B) = prob(A`) * prob(B) by Th22;
hence thesis;
end;
theorem
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
by Th16,XCMPLX_1:6;