Schéma de Bernoulli. Loi binomiale

S S
p=p(S)q=p(S)=1−p
S=
S=p=1
4q=3 4
n
n
S p =p(S)q=p(S) = 1 −p

k06k6n
k n −k S
n
kk n
k n 06k < n n+1
k+1=n k+1+n k
→
n
kn k
n∈N∗p∈[0; 1] X
n p B(n;p)k n
p(X=k) = n
k×pk×qn−kq= 1 −p
p=p(S)q=p(S)=1−p n
X S
X n p
k0
n(X=k)k
S n −k S n
k
pkqn−k
p(X=k) = n
k×pk×qn−k
X n p
XB(n;p)
(X) = np (X) = np(1 −p)

−
−
•
−
−
•
→
S=p=p(S) = 1
6X
XB100; 1
6
p(X= 10) = 100
10 ×1 610 ×5 690 ≈0,0214 ?
p(X>3) = 1 −p(X < 3) = 1 −[p(X= 0) + p(X= 1) + p(X= 2)]
= 1 −100
0×1
60×5 6100 −100 1×1 61×5 699 −100 2×1 62×5 698 ≈0,999997
S=p=p(S) = 1
5= 0,2
X
XB(20; 0,2)
p(X= 15) = 20
15×(0,2)15 ×(0,8)5≈1,7×10−7p(X>18) =
20
X
k=18
20
k×(0,2)k×(0,8)20−k
=20
18×(0,2)18 ×(0,8)2+20 19×(0,2)19 ×(0,8)1+20 20×(0,2)20 ×(0,8)0≈3,3×10−11
(X) = 20 ×0,2 = 4
?•100
10 
•XB(n;p) 0 6k6n p(X=k)
p(X6k)