:: Introduction to Probability
:: by Jan Popio{\l}ek
::
:: Received June 13, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users

theorem Th1: :: RPR_1:1

for E being non empty set
for e being non empty Subset of E holds
( e is Singleton of E iff for Y being set holds
( Y c= e iff ( Y = {} or Y = e ) ) )

proof end;

registration

let E be non empty set ;

cluster 1 -element -> finite for Element of K21(E);

coherence
for b₁ being Singleton of E holds b₁ is finite ;

end;

theorem :: RPR_1:2

for E being non empty set
for A, B being Subset of E
for e being Singleton of E st e = A \/ B & A <> B & not ( A = {} & B = e ) holds
( A = e & B = {} )

proof end;

theorem :: RPR_1:3

for E being non empty set
for A, B being Subset of E
for e being Singleton of E holds
( not e = A \/ B or ( A = e & B = e ) or ( A = e & B = {} ) or ( A = {} & B = e ) )

proof end;

theorem :: RPR_1:4

for E being non empty set
for a being Element of E holds {a} is Singleton of E ;

theorem :: RPR_1:5

for E being non empty set
for e1, e2 being Singleton of E st e1 c= e2 holds
e1 = e2 by Th1;

theorem Th6: :: RPR_1:6

for E being non empty set
for e being Singleton of E ex a being Element of E st
( a in E & e = {a} )

proof end;

theorem :: RPR_1:7

for E being non empty set ex e being Singleton of E st e is Singleton of E

proof end;

theorem :: RPR_1:8

for E being non empty set
for e being Singleton of E ex p being FinSequence st
( p is FinSequence of E & rng p = e & len p = 1 )

proof end;

definition

let E be set ;
mode Event of E is Subset of E;

end;

theorem :: RPR_1:9

for E being non empty set
for e being Singleton of E
for A being Event of E holds
( e misses A or e /\ A = e )

proof end;

theorem :: RPR_1:10

for E being non empty set
for A being Event of E st A <> {} holds
ex e being Singleton of E st e c= A

proof end;

theorem :: RPR_1:11

for E being non empty set
for e being Singleton of E
for A being Event of E holds
( not e c= A \/ (A `) or e c= A or e c= A ` )

proof end;

theorem :: RPR_1:12

for E being non empty set
for e1, e2 being Singleton of E holds
( e1 = e2 or e1 misses e2 )

proof end;

theorem Th13: :: RPR_1:13

for E being non empty set
for A, B being Subset of E holds A /\ B misses A /\ (B `)

proof end;

Lm1: for E being non empty finite set holds 0 < card E

proof end;

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);

coherence
(card A) / (card E) is Real ;

end;

:: deftheorem defines prob RPR_1:def 1 :
for E being finite set
for A being Event of E holds prob A = (card A) / (card E);

theorem :: RPR_1:14

for E being non empty finite set
for e being Singleton of E holds prob e = 1 / (card E) by CARD_1:def 7;

theorem :: RPR_1:15

for E being non empty finite set holds prob ([#] E) = 1 by XCMPLX_1:60;

theorem Th16: :: RPR_1:16

for E being non empty finite set
for A, B being Event of E st A misses B holds
prob (A /\ B) = 0 by CARD_1:27;

theorem :: RPR_1:17

for E being non empty finite set
for A being Event of E holds prob A <= 1

proof end;

theorem Th18: :: RPR_1:18

for E being non empty finite set
for A being Event of E holds 0 <= prob A

proof end;

theorem Th19: :: RPR_1:19

for E being non empty finite set
for A, B being Event of E st A c= B holds
prob A <= prob B

proof end;

theorem Th20: :: RPR_1:20

for E being non empty finite set
for A, B being Event of E holds prob (A \/ B) = ((prob A) + (prob B)) - (prob (A /\ B))

proof end;

theorem Th21: :: RPR_1:21

for E being non empty finite set
for A, B being Event of E st A misses B holds
prob (A \/ B) = (prob A) + (prob B)

proof end;

theorem Th22: :: RPR_1:22

for E being non empty finite set
for A being Event of E holds
( prob A = 1 - (prob (A `)) & prob (A `) = 1 - (prob A) )

proof end;

theorem Th23: :: RPR_1:23

for E being non empty finite set
for A, B being Event of E holds prob (A \ B) = (prob A) - (prob (A /\ B))

proof end;

theorem Th24: :: RPR_1:24

for E being non empty finite set
for A, B being Event of E st B c= A holds
prob (A \ B) = (prob A) - (prob B)

proof end;

theorem :: RPR_1:25

for E being non empty finite set
for A, B being Event of E holds prob (A \/ B) <= (prob A) + (prob B)

proof end;

theorem Th26: :: RPR_1:26

for E being non empty finite set
for A, B being Event of E holds prob A = (prob (A /\ B)) + (prob (A /\ (B `)))

proof end;

theorem :: RPR_1:27

for E being non empty finite set
for A, B being Event of E holds prob A = (prob (A \/ B)) - (prob (B \ A))

proof end;

theorem :: RPR_1:28

for E being non empty finite set
for A, B being Event of E holds (prob A) + (prob ((A `) /\ B)) = (prob B) + (prob ((B `) /\ A))

proof end;

theorem Th29: :: RPR_1:29

for E being non empty finite set
for 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 end;

theorem :: RPR_1:30

for E being non empty finite set
for 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 end;

theorem :: RPR_1:31

for E being non empty finite set
for A, B being Event of E holds (prob A) - (prob B) <= prob (A \ B)

proof end;

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);

coherence
(prob (A /\ B)) / (prob B) is Real ;

end;

:: deftheorem defines prob RPR_1:def 2 :
for E being finite set
for B, A being Event of E holds prob (A,B) = (prob (A /\ B)) / (prob B);

theorem :: RPR_1:32

for E being non empty finite set
for A being Event of E holds prob (A,([#] E)) = prob A

proof end;

theorem :: RPR_1:33

for E being non empty finite set holds prob (([#] E),([#] E)) = 1

proof end;

theorem :: RPR_1:34

for E being non empty finite set
for A, B being Event of E st 0 < prob B holds
prob (A,B) <= 1

proof end;

theorem :: RPR_1:35

for E being non empty finite set
for A, B being Event of E st 0 < prob B holds
0 <= prob (A,B)

proof end;

theorem Th36: :: RPR_1:36

for E being non empty finite set
for A, B being Event of E st 0 < prob B holds
prob (A,B) = 1 - ((prob (B \ A)) / (prob B))

proof end;

theorem :: RPR_1:37

for E being non empty finite set
for A, B being Event of E st 0 < prob B & A c= B holds
prob (A,B) = (prob A) / (prob B)

proof end;

theorem Th38: :: RPR_1:38

for E being non empty finite set
for A, B being Event of E st A misses B holds
prob (A,B) = 0

proof end;

theorem Th39: :: RPR_1:39

for E being non empty finite set
for 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 end;

theorem Th40: :: RPR_1:40

for E being non empty finite set
for 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 end;

theorem Th41: :: RPR_1:41

for E being non empty finite set
for A, B being Event of E st 0 < prob B & B c= A holds
prob (A,B) = 1

proof end;

theorem :: RPR_1:42

for E being non empty finite set
for B being Event of E st 0 < prob B holds
prob (([#] E),B) = 1 by Th41;

theorem :: RPR_1:43

for E being non empty finite set
for A being Event of E holds prob ((A `),A) = 0

proof end;

theorem :: RPR_1:44

for E being non empty finite set
for A being Event of E holds prob (A,(A `)) = 0

proof end;

theorem Th45: :: RPR_1:45

for E being non empty finite set
for A, B being Event of E st 0 < prob B & A misses B holds
prob ((A `),B) = 1

proof end;

theorem Th46: :: RPR_1:46

for E being non empty finite set
for 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 end;

theorem :: RPR_1:47

for E being non empty finite set
for 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 end;

theorem :: RPR_1:48

for E being non empty finite set
for 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 end;

theorem Th49: :: RPR_1:49

for E being non empty finite set
for 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 end;

theorem Th50: :: RPR_1:50

for E being non empty finite set
for 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 end;

theorem Th51: :: RPR_1:51

for E being non empty finite set
for 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 end;

theorem :: RPR_1:52

for E being non empty finite set
for 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 end;

theorem :: RPR_1:53

for E being non empty finite set
for 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 end;

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
for A, B being Event of E st prob (A /\ B) = (prob A) * (prob B) holds
prob (B /\ A) = (prob B) * (prob A) ;

end;

:: deftheorem defines are_independent RPR_1:def 3 :
for E being finite set
for A, B being Event of E holds
( A,B are_independent iff prob (A /\ B) = (prob A) * (prob B) );

theorem :: RPR_1:54

for E being non empty finite set
for A, B being Event of E st 0 < prob B & A,B are_independent holds
prob (A,B) = prob A

proof end;

theorem :: RPR_1:55

for E being non empty finite set
for A, B being Event of E st prob B = 0 holds
A,B are_independent

proof end;

theorem :: RPR_1:56

for E being non empty finite set
for A, B being Event of E st A,B are_independent holds
A ` ,B are_independent

proof end;

theorem :: RPR_1:57

for E being non empty finite set
for A, B being Event of E st A misses B & A,B are_independent & not prob A = 0 holds
prob B = 0 by Th16, XCMPLX_1:6;