Journal of Formalized Mathematics
Volume 2, 1990
University of Bialystok
Copyright (c) 1990 Association of Mizar Users

### Introduction to Probability

by
Jan Popiolek

MML identifier: RPR_1
[ Mizar article, MML identifier index ]

```environ

vocabulary FINSEQ_1, FINSET_1, BOOLE, RELAT_1, PROB_1, SUBSET_1, CARD_1,
ARYTM_3, ARYTM_1, RPR_1, REALSET1;
notation TARSKI, XBOOLE_0, SUBSET_1, XREAL_0, FUNCT_1, DOMAIN_1, REAL_1,
FINSEQ_1, FINSET_1, CARD_1, REALSET1;
constructors DOMAIN_1, REAL_1, NAT_1, REALSET1, XREAL_0, MEMBERED, XBOOLE_0;
clusters FINSET_1, RELSET_1, XREAL_0, MEMBERED, ZFMISC_1, XBOOLE_0;
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;

definition let E be non empty set;
cluster non empty trivial Subset of E;
end;

definition let E;
mode El_ev of E is non empty trivial Subset of E;
end;

theorem :: RPR_1:1
for e being non empty Subset of E holds
e is El_ev of E iff for Y holds (Y c= e iff Y = {} or Y = e);

definition let E;
cluster -> finite El_ev of E;
end;

reserve e, e1, e2 for El_ev of E;

canceled 3;

theorem :: RPR_1:5
e = A \/ B & A <> B implies
A = {} & B = e or A = e & B = {};

theorem :: RPR_1:6
e = A \/ B implies A = e & B = e or
A = e & B = {} or A = {} & B = e;

theorem :: RPR_1:7
{a} is El_ev of E;

canceled 2;

theorem :: RPR_1:10
e1 c= e2 implies e1 = e2;

theorem :: RPR_1:11
ex a st a in E & e = {a};

theorem :: RPR_1:12
ex e st e is El_ev of E;

canceled;

theorem :: RPR_1:14
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;

canceled 7;

theorem :: RPR_1:22
for E being non empty set, e being El_ev of E, A being Event of E
holds e misses A or e /\ A = e;

canceled 2;

theorem :: RPR_1:25
for E being non empty set, A being Event of E st A <> {}
ex e being El_ev of E st e c= A;

theorem :: RPR_1:26
for E being non empty set, e being El_ev of E, A being Event of E
st e c= A \/ A` holds e c= A or e c= A`;

theorem :: RPR_1:27
e1 = e2 or e1 misses e2;

canceled 6;

theorem :: RPR_1:34
A /\ B misses A /\ B`;

definition
let E be finite non empty set;
let A be Event of E;
canceled 3;

func prob(A) -> Real equals
:: RPR_1:def 4
card A / card E;
end;

canceled 3;

theorem :: RPR_1:38
for E being finite non empty set, e being El_ev of E holds
prob(e) = 1 / card E;

theorem :: RPR_1:39
for E being finite non empty set holds prob([#] E) = 1;

theorem :: RPR_1:40
for E being finite non empty set holds prob({} E) = 0;

theorem :: RPR_1:41
for E being finite non empty set, A,B being Event of E st A misses B
holds prob(A /\ B) = 0;

theorem :: RPR_1:42
for E being finite non empty set, A being Event of E holds prob(A) <= 1;

theorem :: RPR_1:43
for E being finite non empty set, A being Event of E holds 0 <= prob(A);

theorem :: RPR_1:44
for E being finite non empty set, A,B being Event of E st A c= B holds
prob(A) <= prob(B);

canceled;

theorem :: RPR_1:46
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_1:47
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_1:48
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_1:49
for E being finite non empty set, A,B being Event of E holds
prob(A \ B) = prob(A) - prob(A /\ B);

theorem :: RPR_1:50
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_1:51
for E being finite non empty set, A,B being Event of E holds
prob(A \/ B) <= prob(A) + prob(B);

canceled;

theorem :: RPR_1:53
for E being finite non empty set, A,B being Event of E holds
prob(A) = prob(A /\ B) + prob(A /\ B`);

theorem :: RPR_1:54
for E being finite non empty set, A,B being Event of E holds
prob(A) = prob(A \/ B) - prob(B \ A);

theorem :: RPR_1:55
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_1:56
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_1:57
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_1:58
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 non empty set;
let B,A be Event of E;
func prob(A , B) -> Real equals
:: RPR_1:def 5
prob(A /\ B) / prob(B);
end;

canceled;

theorem :: RPR_1:60
for E being finite non empty set, A,B being Event of E st 0 < prob(B)
holds prob(A /\ B) = prob(A , B) * prob(B);

theorem :: RPR_1:61
for E being finite non empty set, A being Event of E holds
prob(A , [#] E ) = prob(A);

theorem :: RPR_1:62
for E being finite non empty set holds prob([#] E , [#] E) = 1;

theorem :: RPR_1:63
for E being finite non empty set holds prob({} E , [#] E) = 0;

theorem :: RPR_1:64
for E being finite non empty set, A,B being Event of E st 0 < prob(B)
holds prob(A , B) <= 1;

theorem :: RPR_1:65
for E being finite non empty set, A,B being Event of E st 0 < prob(B)
holds 0 <= prob(A , B);

theorem :: RPR_1:66
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_1:67
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_1:68
for E being finite non empty set, A,B being Event of E st
A misses B holds prob(A , B) = 0;

theorem :: RPR_1:69
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_1:70
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_1:71
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_1:72
for E being finite non empty set, B being Event of E st 0 < prob(B)
holds prob([#] E , B) = 1;

theorem :: RPR_1:73
for E being finite non empty set, A being Event of E st 0 < prob(A)
holds prob(A` , A) = 0;

theorem :: RPR_1:74
for E being finite non empty set, A being Event of E st prob(A) < 1
holds prob(A , A`) = 0;

theorem :: RPR_1:75
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_1:76
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_1:77
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_1:78
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_1:79
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_1:80
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_1:81
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_1:82
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_1:83
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 non empty set;
let A, B be Event of E;
pred A, B are_independent means
:: RPR_1:def 6
prob(A /\ B) = prob(A) * prob(B);
symmetry;
end;

canceled 2;

theorem :: RPR_1:86
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_1:87
for E being finite non empty set, A,B being Event of E st
prob(B) = 0 holds A , B are_independent;

theorem :: RPR_1:88
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_1:89
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;
```