Mathématiques générales (été) Corrigé de la série 4 Exercice 1 :

A B
P(B|A)
B
A
N(nr)
N(rr)
N(nr)=4·8 = 32 N(rr) = 8 ·7 = 56.
P(B|A) = N(nr)
N(nr) + N(rr)=4 11.
Rjj
Njj j = 1,2
P(N1|R2) P(N1) = 4
12 =1 3P(R2|R1) = 7 11
P(R2|N1) = 8
11 N1R1
P(N1|R2) = P(R2|N1)P(N1)
R2|N1)P(N1)+R2|R1)P(R1)
=
8
11
1
3
8
11
1
3+7 11
2
3
=4
11.
Ann
P(A1\A2\A3) = P(A1)·P(A2|A1)·P(A3|(A1\A2)).
C15
5
3·C12
4
P(A1) = 3·C12
4
C15
5
=45
91.
P(A2|A1)

P(A2|A1) = 2·C8
4
C10
5
=5
9.
P=25
91
Ω = {(P, P, P ),(P, P, F ),(P, F, P ),(P, F, F ),(F, P, P ),(F, P, F ),(F, F, P ),(F, F, F )}.
1
8
P(Gi) = 1
2i= 1,2,3
P(G1∩G2)G1G2
P(G1∩G2) = 1
4=P(G1)·P(G2)
G1G2
G1∩G2∩G3
G1∩G2
P(G1∩G2∩G3) = P(G1∩G2) = 1
46=1 8=P(G1)·P(G2)·P(G3)
P(B|A) = P(A∩B)
P(A)⇒P(A∩B) = P(B|A)·P(A)
P(A∩B) = 1
8
P(A|B) = P(A∩B)
P(B)⇒P(B) = P(A∩B) P(A|B)
P(B) = 3
8
P(A∪B) = P(A) + P(B)−P(A∩B) = 1
2
A B
P(B|A)
•
1
2

•
1
2
P(A∩B) = P(B)P(A |B) = 1
2·1 4=1 8
•
P(A) = 1
2+1 8=5 8
P(B|A) = P(A∩B)
P(A)=1 5
A B
P(A|B)P(B|A)=0.95 P(B|¯ A)=0.01 ¯ A A
P(A|B)
P(A)P(¯ A)
P(A) = 2
40 =1 20 P(¯
A) = 19
20
P(A|B) = P(B|A)P(A)
P(B|A)P(A) + P(B|¯ A)P(¯ A)=5
6
P(B∩C) = P(B)·P(C)B C
P(B∪C) = P(B)+P(C)−P(B∩C)A B ∪C
P(A) + P(B∪C) = 1 P(B) = x
x
2+x+x−x2= 1 ⇔x2−5 2x+ 1 = 0 ⇔x= 2 1 2
1
2
P(A) = 1
4