第六次课讲义1
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2.1 ^‡VÇ 31
e¯‡ A
1
u), Kl1˜‡‡f¥r m + n − 1 ‡¥þØ´1 1 Òx¥, ^¯‡ B
j
L«1 j
gl1˜‡‡fpr¥Ø´1 1 Òx¥, = A
1
= B
1
B
2
· · · B
m+n−1
. Šâ¦{úªk
P (A
1
) = P (B
1
)P (B
2
|B
1
) · · · P (B
m
|B
1
B
2
· · · B
m−1
) × P (B
m+1
B
m+2
· · · B
m+n−1
|B
1
B
2
· · · B
m
)
=
1 −
1
n
m
P (B
m+1
B
m+2
· · · B
m+n−1
|B
1
B
2
· · · B
m
) =
1 −
1
n
m
×
1
n
.
ddŒ•1˜‡‡f•˜gr¥´x¥VÇ• (1 − 1/n)
m
.
2.1.2 VÇúª
|^^‡VÇŒ±ò˜‡E,¯‡V ÇOŽ¯K?1{z, ùÒ´!¤ù VVVÇÇÇúúúªªª,
´VÇØ¥•Äúªƒ˜. ÙŸ´é\{Ú¦{¯‡nÜ$^: é?¿p؃N¯‡
A, B k P (A ∪ B) = P (A) + P (B); é?¿¯‡ A, B ÷v P (A) > 0 k P (AB) = P (A)P (B|A).
Äk½Â˜m˜‡y©.
½Â 2.2 e‘ů‡ A
1
, A
2
, · · · , A
n
÷v: i) ?¿üü¯‡´p؃N5 (½p½), =
A
i
∩ A
j
= ∅ (i 6= j); ii) 5 Ω =
S
n
i=1
A
i
, K¡¯‡ A
1
, A
2
, · · · , A
n
•˜m Ω ˜‡ yyy©©©.
AO/, n = 2 ž k A
1
=
¯
A
2
, = A
1
† A
2
p• éᯇ. e
A
1
, A
2
, · · · , A
n
•˜m˜‡y©, KzgÁž¯‡ A
1
, A
2
, · · · , A
n
k…=k˜‡¯‡u).
Äu˜my©, e¡0VÇúª:
½n 2.1 e¯‡ A
1
, A
2
, · · · , A
n
•˜m Ω ˜‡y©, é?¿¯‡ B k
P (B) =
n
X
i=1
P (BA
i
) =
n
X
i=1
P (A
i
)P (B|A
i
),
¡ƒ• VVVÇÇÇúúúªªª (Law of total probability).
Œ±ò¯‡ B wŠ,˜L§(J, ò A
1
, A
2
, · · · , A
n
wŠ)T (
JeZÏ. e i) z˜«Ï®•, = P (A) ®•; ii) z˜«Ïé(J
B K•®•, = P (B|A
k
) ®•, K P (B) ŒOŽ.
y² Šâ©Æk
B = B ∩ Ω = B
\
n
[
i=1
A
i
!
=
n
[
i=1
BA
i
d A
i
∩ A
j
= ∅ Œ BA
i
∩ BA
j
= ∅, dVÇk•ŒŒ\5k
P (B) = P
n
[
i=1
BA
i
!
=
n
X
i=1
P (BA
i
) =
n
X
i=1
P (A
i
)P (B|A
i
).
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