:: Introduction to Probability
:: by Jan Popio{\l}ek
::
:: Received June 13, 1990
:: Copyright (c) 1990-2016 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;