Ch1 1
a0a2a1a2a3 a4a2a5a2a6a2a7a2a8a2a6a2a9a2a10a2a11a2a12a2a13a15a14a2a16
x1 a4a2a5a2a6a2a7
1,a17a2a18a2a19a2a20a2a21
a6a2a9
a22a24a23a24a25a24a26a24a27a24a28a24a29a2a30a32a31a34a33a2a35a32a36a38a37a2a39a2a33a2a35a2a40a32a36a34a37a2a41a2a42a2a43a45a44a47a46a2a48a2a49a2a50a2a27a2a51a2a52a2a53
a54a2a55a2a56a2a22a2a23a2a25a2a26a2a57a15a58a2a41a15a33a15a35a15a59a2a41a15a60a15a61a45a62a63a25a15a26 Aa57a2a58a2a41a2a48a2a49a2a64a2a65 P(A)a44a67a66a34a68
a69a2a70 06 P(A) 6 1a62a72a71a2a73a2a74a2a75a76a44 P(;)=0.
a77a2a78a2a25a2a26a76a44a79a27a2a40a2a35a2a80a2a81a2a82a2a41a15a25a2a26a45a62a72a25a15a26a2a33a15a83a2a84a15a85a2a65a15a86a2a87a15a77a2a78a15a25a2a26a15a88a2a89a15a41
a75a2a90a76a62a72a91a2a92a76a44a72a93a15a94a15a95a45a44 A
i=f
a36a34a37 iga44 i =1a44 2a44 3a44 4a44 5a44 6a50a2a27a2a77a2a78a2a25a2a26a76a62
a96 a78a45a44a63a50a2a27a2a97a2a98a2a99a15a100a2a29a15a30a15a33a15a35a101a36a102a37a15a41a2a42a15a103a104a62 a96 a78a15a74a101a105a106a44a107a65a15a108 a70a15a96 a78a2a109a15a110
a111a24a108a24a88a24a89a24a41a2a75a15a90a45a44a112a64a2a65 a62a114a113a24a115a24a41a24a116a2a117 a70 P( ) = 1a62a114a83a24a118a24a116a24a117a24a119a15a22a2a23a15a25a2a26a2a120
a121 a65 a96 a78a2a109a2a41a2a86a2a122a2a75a2a90a45a44a123a25a15a26a15a41a15a124a2a125a15a50a2a27a15a75a15a90a2a41a15a124a15a125a45a62
a71a2a73a2a48a2a49 Pa44a72a126a2a127a2a128a2a129a2a83a2a118a2a130a2a122a15a131a2a26a104a132
(1)P(A + B) = P(A) + P(B) ;
(2)P(Ac) = 1 P(A);
(3)A B =) P(A) 6 P(B);
2,a4a2a5a2a6a2a7
a133a2a134a2a135 a41 a96 a78a2a74a32a105a34a27a2a136a2a137 a70a2a138 a41a15a75a2a90a45a62a72a46a15a71a2a139a15a97a2a122a15a77a2a78a15a25a2a26a45a44 a133a15a134a2a135 a41
a140a2a141a2a50a2a27a2a142a2a117a2a57a2a58a2a41a2a48a15a49a15a143a2a144a15a145a45a62a123a73a15a27a45a44a123a116a15a117a15a146a2a147a15a148a2a149a15a150a15a48a2a151a15a41a2a52a15a152a15a153a2a154a104a132
(1) = f!1; !2; ; !ng;
(2)P(!k) = 1=n; k = 1; 2; ; n;
a149a2a150a2a48a2a151a2a27a2a155a2a156 a96 a78a2a74a32a105a102a41 a133a15a157a15a158 a41a15a48a2a49a15a159a15a151a45a62
a160 1.1 a93a2a161a2a97a2a162a2a163a2a164a2a41a2a165a2a166a76a44a123a25a15a26 A = f
a167a34a168a2a169
a69 ga44a72a170 P(A)a62
a171 a127a15a29a15a30a15a41a15a42a15a103 a70a173a172 a122a32a132
a167a174a168a175a169
a69a175a176
a167a174a168a175a169
a118a177a44a178a142a175a117a175a27a175a145a173a33a175a35a175a41a177a62a178a73
a27a76a44
P(A) = 1=2:
a160 1.2
a179
a30 100a26a2a180a2a181a45a44a107a182a32a31 a70 5a26a2a27a2a98a2a181a45a62a107a183a2a184 50a26a45a44a107a170a2a182a32a31a34a185 a70
1
Ch1 2
a172 a26a2a27a2a98a2a181a2a41a2a48a2a49a76a62
a171a187a186 a31a174a184a188a36 50 a26a175a189 a70 C50
100 a190a175a191a175a192
a44a102a182a188a31a174a185 a70 2 a26a175a98a175a181a175a41a175a184
a192
a189 a70
C4895C25
a190
a44a72a73a2a27a76a44a72a184a32a36 50a26a2a185 a70a2a172 a26a2a27a2a98a2a181a2a41a2a48a2a49a15a65
C50100
C4895C25 =
8
25:
a160 1.3
a193
a58a195a194a34a196a34a197a188a198a79a97a15a122a15a199 a69a15a70 n a122a2a152a2a58a45a44 A = f
a200a2a201
a70a2a172 a122a2a202a2a58a203a194a67a144
a204 ga44a72a170a2a25a2a26 Aa57a2a58a2a41a2a48a2a49a76a62
a171a206a205a2a207 a170 Aa57a2a58a2a41a2a48a2a49a2a208a2a40a2a209a15a210a104a44a63a108a15a83a15a116a15a117a15a211a2a212a15a213 Aa41a2a214a2a75 Ac a57a2a58
a41a2a48a2a49a76a62
P(A) = 1 P(Ac) = 1 A
n
365
365n:
a215 n=40a44 45a44 50a44 55
a216
a44a107a83 a69 a48a2a49a2a82a15a217a15a145a15a73 0.89a44 0.94a44 0.97a44 0.99a62a107a218a2a97a2a42
a103a2a219 a205 a99a2a41a2a220a2a221a2a208a2a40a2a144a15a222a45a62
a149a2a150a2a48a2a151a32a31 a70 a97a2a223a101a196a102a197a45a44a107a50a15a27a15a224a15a225a101a196a34a197a104a62 a134a15a135 a41a15a226a104a44a63a50a15a27a15a97a15a122a15a209a15a227a32a31 a70
a a122a2a228a2a225a45a44 b a122a2a229a2a225a45a44a107a139a2a98 a186 a209a15a227a101a31a102a224a101a36a102a97a15a122a15a225a104a44a107a170 na98a32a31a34a185a2a230 a70 k a122a2a228a2a225
a41a2a48a2a49 p
k
a62
a71a2a231a76a44a72a126a2a127a2a82a2a65 a172
a190
a140a2a232a104a132 a70a15a233a101a234a102a176a2a235a15a233a101a234 a62
a70a2a233a32a234
a216
a44a236a139a2a98a175a41a15a140a15a232a175a143a15a144 a204 a44 n a98a32a31 a70 k a98a2a224a2a147a2a228a2a225a45a44 n k a98a2a224a2a147
a229a2a225a76a62a72a73a2a27 a70
pk = Ckn a
kbn k
(a + b)n = C
k
n
a
a + b
k b
a + b
n k:
a235a2a233a32a234
a216
a44 a a122a2a228a2a225a32a31a34a184a32a36 k a122a76a44 b a122a2a229a2a225a32a31a34a184a32a36 n k a122a76a44a72a73a2a27
pk = C
k
aC
n k
b
Cna+b,
a69a2a237 a41a2a91 1.2a50a2a238a2a73 a235a2a233a32a234 a41a2a224a2a225a32a196a34a197a45a62
x2 a239a2a240
a6a2a7
a149a2a150a2a48a2a151a2a241a2a51a2a145a2a48a2a49a76a44a79a89a2a242a15a41a2a243a15a125a2a148 a96 a78a15a74a32a105a102a65 a70a15a138 a41a15a97a2a87a101a196a34a197a45a62a72a244a15a27
a215a15a96 a78a15a74a101a105a102a65 a235a175a138
a216
a44a245a149a175a150a175a48a175a151a175a50a175a40a175a35a175a246a175a247a175a148a177a62a245a73a175a27a177a44a245a116a175a117a175a248a175a249a175a130a175a250a175a48a175a151a177a44
a251a2a65a2a149a2a150a2a48a2a151a2a28 a235a2a138 a140a15a141a15a41a2a252a15a253a45a62
2
Ch1 3
a211a2a254a2a118 a237 a41a2a97a2a122a2a91a2a95a76a62
a160 2.1
a193a255
a1 a196a38a197a188a198 a172 a202a2a144
a255
a86a3a2 5a109
a200 6a109a24a28a24a86a5a4 a1a15a237 a44a72a211a2a147a7a6a15a145a7a8
a9 a97a2a202 20a82a7a10a76a44a12a11
a216
a50a2a155a7a13a45a62a123a170a15a218 a172 a122a15a202a15a35 a1a15a237 a41a15a48a15a49a45a62
a171 a83 xa44 ya82a24a217a24a84a24a85 a172 a202a24a147a5a14a24a41
a216
a55a76a44a16a15 a1a24a237 a41a7a17a5a18a2a131a2a26a24a27 jx yj 6 20a62
a64
G = f(x; y)j0 6 x 6 60; 0 6 y 6 60g;
g = f(x; y)jjx yj 6 20; (x; y) 2 Gg:
a19 a158a21a20 1-1a44a72a33a7a22 a172 a202a2a35 a1a2a237 a41a2a48a2a49a15a65
P(A) = S(g)S(G) = 60
2 402
602 =
5
9:
y
60
20
O
g
20 60 x
a20 1.1
a28a15a218a3a23a45a44 a116a15a117a3a24a3a25 a69a27a26 a120a15a148 a172 a202a15a147a27a14a175a41
a216
a105a174a28 5 a109
a200 6
a109a3a28a101a105a102a41a15a23 a1
a163a2a145a45a44 a46a7a29a7a30a15a40a3a31a3a32a104a62 a71a175a73a15a97a3a33a15a41a15a140a15a141a104a44 a116a175a117a15a143a15a33a15a83a15a51 a20 a141a175a84a15a85a101a36 a44a35a34a15a110a15a111
a29a24a30a24a42a24a103a24a41 a20 a141a2a84a2a85a2a65 Ga44a36a29a24a183a24a250 a172 a109a24a41a24a23 a1 a163a2a145a45a44 a46 a70 a241a2a73a2a25a2a26 Aa41a24a29a24a30
a42a2a103a2a41 a20 a141a2a84a2a85a2a65 ga44a12a15a2a25a2a26 Aa57a2a58a2a41a2a48a2a49a2a65
P(A) = S(g)S(G):
a218a7a23a2a41 S( )a33a2a83a2a84a2a85 a20 a141a2a41 a237a7a37 a44a123a39a15a33a15a83a15a84a2a85a3a38a7a39a45a44a63a111 a37 a145a45a62
a118 a237 a80a7a40a32a36a34a97a2a122a2a91a2a95a76a62
a160 2.2
a193 Buffon a196a34a197a188a198a42a41 a237a2a69 a56a3a43a15a97a15a87a3a41a3a44a3a45a104a44a107a142a15a117a3a28a101a105a102a41a3a46a2a155a15a143a15a145
a73 aa44a48a47a34a231a7a41 a237a2a69 a183a3a49a15a41a15a93a15a161a15a97a3a38a3a39a15a65 l(l 6 a) a41a7a50a45a44a107a170a2a231a7a50a2a219a15a86a15a97a3a41a3a44a3a45
a144a7a51a2a41a2a48a2a49a76a62
3
Ch1 4
a
’ x
l
x
a
2
O
g
l
2 sin ’
’
a20 1.2
a171 a83 x a84a2a85a7a50a2a41a32a31a102a109a15a147 a133a3a52 a41a175a97a15a131a3a41a27a44a3a45a175a41a3a46a15a155a104a44 ’ a84a2a85a2a127a7a41a7a44a3a45a15a219
a50a2a41a7a53a7a54
a193
a158a21a20 1-2a198a44a72a113a2a115 a70
0 6 x 6 a2; 0 6 ’ 6 ;
a55 a231a76a44
G = f(’; x)j0 6 x 6 a2; 0 6 ’ 6 g;
a46 a215 a50a2a219a7a41a7a44a7a45a2a144a7a51
a216
a44 a70 x 6 l
2 sin ’
a44a72a108a2a83 a70
g = f(’; x)jx 6 l2 sin ’; (’; x) 2 Gg:
a218 a96 a50a2a33a2a83a2a146a2a147
P(A) = S(g)S(G) =
R
0
l
2 sin ’d’
a
2
= 2 la,
a28a2a254 a207 a118a2a54a2a41a2a218a2a122a32a196a34a197a76a62
a160 3.3
a193 Berfrand a56a7a57
a198a112a28a7a58a7a59a2a65 1a41a21a60a7a61a34a22a2a23a2a41a15a184a15a97a15a131a3a62a104a44a178a196a34a182a3a38
a39a7a63a7a11a2a127a21a60a7a61 a207 a145a7a64a7a65a3a54a15a141a2a41a3a64a7a38 p3a41a2a48a2a49a2a27a7a66
a201a68a67
a218a15a27a15a97a15a122a15a130a15a250a15a48a175a151a175a41a188a196a174a197a177a44a245a244a175a27a175a71a70a69 a22a175a23a27a4a21a71a245a97a3a72a175a41a27a73 a121 a41a173a40 a204 a246a27a74a177a44
a33a2a83a2a146a2a147a2a40 a204 a41a7a75a7a76a45a62a123a118 a237 a40a101a36a102a182a32a31a102a41a7a65
a190
a62
a171 1 a183a2a250a7a62a2a143 a176 a60a78a77a27a51a15a73 a172 a109a104a62a236a40a3a79a15a97a27a33a15a59a104a44a236a211a68a80a102a120a15a182a101a31a102a97a175a109a15a28a68a60a78a77
a69 a44a63a83a2a231a2a109a2a65a2a97a7a81a15a109a15a251a68a60a102a41a68a61 a207 a145a3a64a3a65a3a54a15a141a104a44a63a119a68a60a78a77a15a82a15a65a3a65a7a82a15a145a3a38a15a41a3a83a104a62a63a113
a115a76a44a85a84 a70a2a215 a62a2a41 a9 a97a7a81a15a109a7a86a15a28a2a219a15a127a7a81a15a109a2a40a15a144a7a87a15a41a2a97a3a82a7a83 a69 a41
a216
a8a45a44a12a62a3a38a7a88 a1
a63a7a11 p3a62a12a89a2a108a2a170a2a41a2a48a2a49a2a65 1
3
a62 a158a21a20 1-3a62
4
Ch1 5
a171 2 a62a7a38a7a84a7a90a2a142a2a41a3a62a27a91a3a46 a70a3a92 a44a236a219
a191
a47 a235a3a92 a44a236a73a15a27a15a33a15a83 a26a3a93 a142a3a94 a205 a73
a86a24a97a24a131 a205 a59a76a62 a71a24a73a2a218 a96 a41a5a62a45a44 a182a7a38a7a39a24a60a2a73 p3a215 a29a5a95 a215 a142a24a41a5a62a5a91a7a46a2a61a2a73 1
2
a62 a55
a231a2a108a2a170a2a41a2a48a2a49a2a65 1
2
a62
a171 3 a62a7a96a2a182a32a31a34a109a7a97a2a97a2a247a2a120a45a44a98a89 a215 a29a7a95 a215 a182a32a31a102a109a15a238a2a73a3a58a3a59a2a65 1
2
a41 a204 a91a21a60
a216
a44a12a62a7a38a7a88a2a60a2a73 p3a62a72a46a2a231a2a61a21a60 a237a7a37 a65a2a60a68a60 a237a3a37 a41 1
4
a44a12a89a2a108a2a170a2a41a2a48a2a49a2a65 1
4
a62
(a) (b) (c)
d
a20 1.3
a204 a97a32a196a34a197a2a146a2a147a7a65
a190
a40 a204 a41a3a75a3a76a104a44a100a99 a55 a27a3a101a15a51a15a148a15a40 a204 a41a15a145a2a33a15a35a15a59a104a62a107a28a15a246 1
a31 a44 a26 a120a24a97a7a102a21a80a102a120a2a46 a9 a97a7a102a15a28a21a60a103a77 a69 a163a15a164a2a82a7a104a106a105a112a246 2 a31 a26a5a93 a62a24a41a32a31a34a109a2a28a15a97a21a80
a120 a205 a59 a69 a163a2a164a2a82a7a104a107a105a123a46a15a246 3 a15 a26 a120a7a62a2a41a32a31a34a109a2a28a21a60a3a61a34a163a15a164a15a82a3a104a45a62 a55 a231a45a44a108a65
a190
a40
a204 a41a3a75a3a76a15a27a3a50a15a71a3a65
a190
a40 a204 a41a15a22a175a23a175a29a15a30
a193a110a109a27a111
a254a3a112a175a54a175a144 a204 a198a44 a71a175a73a3a113a115a114 a41a15a22a175a23
a29a2a30a2a46a7a116a76a44a72a142a2a117a2a143a15a27a3a117a7a118a15a41a45a62
a55 a231a2a28a7a119a2a51a120a69 a22a2a23a107a71a122a121a123a69 a183a15a184a107a71a122a121a123a69 a145a15a33a2a35a106a71a124a121a123a69 a163a15a164a106a71
a216
a44a123a126a15a127a21a125a78a118a15a182a7a73 a121 a62
x3 a6a2a9a7a126a7a127a2a8a2a6a2a9a2a10a2a11a2a13a15a14a2a16
a128 a172a5a129 a40a177a36a38a148a24a48a2a49 a205 a99 a69 a41a24a120 a121 a44 a207 a118a2a54a2a119a7a130a7a131a7a118a2a41a2a153a2a154a2a48a2a49a7a132a7a39 a176 a48
a49a2a74a32a105 a62
a128 a237a21a133a103a134a7a135 a11a76a44a79a25a2a26a2a50a2a27a2a86a15a87 a96 a78a2a109a15a88a2a89a15a41a2a75a15a90a45a44a72a25a2a26a3a28a32a105a102a41a2a124a15a125a2a39a15a50
a27a24a75a24a90a24a124a24a125a45a62a72a244a2a27a104a44a79a208a7a136 a70 a71a15a75a2a90a15a249a7a44 a138a7a137 a62a72a71a2a73a15a25a2a26a45a44a72a97a2a122a3a138a21a125a102a113a2a41a3a18a2a170a45a44
a50a24a27a24a108 a70 a25a2a26a15a88a2a89a2a41a15a75a2a90a2a71a15a73a2a208
a193 A [Ba198a139a121a42a51 a193 A \B a140
a64a24a65 ABa198a139a121a42a141
a193 Aca198
a145a5a65
a190
a124a24a125a5a142a21a143a106a62a144a128a7a145a7a146a2a152a3a6a7a147a7a148a7a149a7a150a7a151a7a152a2a73 1933
a153
a28a155a154 a48a24a49
a57
a77a7a156a2a48a7a157a107a158
a97a7a159a32a31 a44a79a51a7a160a7a161a7a162a2a41
a191a15a192
a219a15a75a2a90
a57
a41a15a99a2a109a15a89a2a242a3a4a2a246a15a247a2a148a15a218a2a122a101a196a34a197a45a44a12a163a101a36a34a148a15a48
a49a2a74a32a105a34a41a2a48a7a157a76a62
1 a6a2a9a7a126a7a127a7a164a7a165a7a166a7a167a7a168
(1)a96 a78a2a74a32a105
5
Ch1 6
a27a2a97a7a169a2a74a2a75a2a90a76a44a72a65 a96 a78a2a74a101a105a106a62
(2) F a219a2a33a7a132a2a74a32a105
F a27a21a170 a41a2a86a2a87a2a95a2a75a2a108a7a171a2a89a2a41a2a97a2a122 a172
a52a76a44a12a173a2a142a2a128a2a129a45a132
(1) 2 F ;
(2) a34 A 2 Fa44a12a15 Ac 2 F;
(3) a34 An 2 Fa44 n =1a44 2a44 a44a12a15
1[
n=1
An 2 F;
a174 ( a44 F)a65a2a33a7a132a2a74a32a105 a62
(3) P a219a2a48a2a49a2a74a32a105
a48a2a49 P a27a2a120 a121 a28 F a69 a41a7a175a2a52a76a44a12a173a2a142a2a27a15a97a2a122 a186 F a147 [0; 1]a41a7a176a7a177a45a132
P,F ! [0; 1]a44a12a29a2a142a2a128a2a129a45a132
(1) 8A 2 Fa44 P(A) > 0 ;
(2) P( ) = 1;
(3) a34 An 2 Fa44 n =1a44 2a44 a44a12a29 An a30a2a40a2a144a7a51a76a44a12a15
P(
1[
n=1
An) =
1X
n=1
P(An);
a174 a218 a96 a41 Pa65 a33a178a132 a74a104a105 ( ;F)a69 a41 a97 a122 a48 a49a178a132a178a39
a193
a134 a174a24a65a24a48 a49a101a198a44a179a174 ( ;F; P)
a65a2a97a2a48a2a49a2a74a32a105 a62
a33a2a83a2a252a32a36 a44a72a48a2a49a7a132a7a39 P a70 a83a2a118a2a59a7a180a45a132
(1) P(;) = 0 ;
(2) a34 Ak 2 Fa44 k =1a44 2a44 a44 na44a12a29 Ak a30a2a40a2a144a7a51a76a44a12a15
P(
n[
k=1
Ak) =
nX
k=1
P(Ak);
(3) a34 P(Ac) = 1 P(A);
(4) P(A B) = P(A) P(B) ;
(5) a34 A 2 Ba44a12a15 P(A) 6 P(B) ;
(6) P(A[ B) 6 P(A) + P(B) ;
(7) P(Snk=1 Ak) 6 Pnk=1 P(Ak) ;
6
Ch1 7
(8) a34 A1 2 A2 2 a44a12a15
P(
1[
n=1
) = lim
n!1
P(An);
2 a181a7a182
a13a7a183
a170 a83 a69 a48 a49 a41 a59a178a180 a33a24a83 a146a24a147a5a184
a192
a160a5a185a24a41 a134a24a135 a140a24a141a76a132 a71 a183a5a49 a172 a122a24a25a24a26 Aa44 Ba44
a70
P(A[ B) = P(A) + P(B) P(AB):
a134 a18a7a186a21a125a34a92a15a118a177a132a187a170 A [ B = A [ BAca44 B = BA [ BAca44a35a29 A a219 BAc a40a2a144
a51a76a44 BA a219 BAc a40a2a144a7a51a76a44 a70
P(A [ B) = P(A) + P(BAc);
P(BAc) = P(B) P(BA);
a172 a185a2a144a7a184a7a173a2a146a2a108a7a18a2a41a7a160a3a185a45a62
a71a2a73a7a66a2a122a2a75a2a90a2a41a2a140a2a141a76a44a123a116a15a117 a70 a118 a237 a41 Jordan a160a7a185a45a132
P(
n[
i=1
Ai) =
nX
k=1
( 1)k+1
X
16i1<i2< <ik6n
P(Ai1Ai2 Aik):
a51a2a209a7a188a7a99a7a161
a140
a6a2a52a2a152a7a189a3a190
a192
a163a15a33a7a40a101a36a78a186a21a125a106a62
a118 a237 a40a32a36a34a97a2a122a2a91a2a95a76a62
a160 3.1
a193a192a191
a71a32a196a34a197a188a198 n a71a7a152a7a193a2a22a2a23a7a194a7a195a7a196a15a28a3a38a3a197a15a41 a172a3a198 a44a107a211a15a58a15a110a7a199a3a196a15a28
a182a32a31a34a41a2a97 a198 a44 a196
a200a15a201
a70 a97a15a71a7a152a3a193 a237 a71 a237 a196a15a41a15a48a2a49a15a27a3a66
a201a155a67
a171 a64
B =a200a2a201
a70 a97a2a71a7a152a7a193 a237 a71 a237 a196a45a44
Ai= a200 ia71a7a152a7a193 a237 a71 a237 a196a76a44 i =1a44 2a44a103a201 a44 na62
a210a7a22 B = Sn
i=1 Ai
a62a72a71a2a183a7a49a2a41 1 6 i
1 < i2 < < ik 6 n
a44 a70
P(Ai1Ai2) Aik = (n k)!n! ;
X
16i1<i2< <ik6n
P(Ai1Ai2 Aik) = Ckn (n k)!n! = 1k!;
7
Ch1 8
a73a2a27a76a44a202a170 Jordan a160a7a185a2a33a7a22
P(B) =
nX
k=1
( 1)k+1 1k!:
a215 n ! 1
a216
a44 P(B) =
nX
k=1
( 1)k+1 1k! ! 1 e 1a62
8
a0a2a1a2a3 a4a2a5a2a6a2a7a2a8a2a6a2a9a2a10a2a11a2a12a2a13a15a14a2a16
x1 a4a2a5a2a6a2a7
1,a17a2a18a2a19a2a20a2a21
a6a2a9
a22a24a23a24a25a24a26a24a27a24a28a24a29a2a30a32a31a34a33a2a35a32a36a38a37a2a39a2a33a2a35a2a40a32a36a34a37a2a41a2a42a2a43a45a44a47a46a2a48a2a49a2a50a2a27a2a51a2a52a2a53
a54a2a55a2a56a2a22a2a23a2a25a2a26a2a57a15a58a2a41a15a33a15a35a15a59a2a41a15a60a15a61a45a62a63a25a15a26 Aa57a2a58a2a41a2a48a2a49a2a64a2a65 P(A)a44a67a66a34a68
a69a2a70 06 P(A) 6 1a62a72a71a2a73a2a74a2a75a76a44 P(;)=0.
a77a2a78a2a25a2a26a76a44a79a27a2a40a2a35a2a80a2a81a2a82a2a41a15a25a2a26a45a62a72a25a15a26a2a33a15a83a2a84a15a85a2a65a15a86a2a87a15a77a2a78a15a25a2a26a15a88a2a89a15a41
a75a2a90a76a62a72a91a2a92a76a44a72a93a15a94a15a95a45a44 A
i=f
a36a34a37 iga44 i =1a44 2a44 3a44 4a44 5a44 6a50a2a27a2a77a2a78a2a25a2a26a76a62
a96 a78a45a44a63a50a2a27a2a97a2a98a2a99a15a100a2a29a15a30a15a33a15a35a101a36a102a37a15a41a2a42a15a103a104a62 a96 a78a15a74a101a105a106a44a107a65a15a108 a70a15a96 a78a2a109a15a110
a111a24a108a24a88a24a89a24a41a2a75a15a90a45a44a112a64a2a65 a62a114a113a24a115a24a41a24a116a2a117 a70 P( ) = 1a62a114a83a24a118a24a116a24a117a24a119a15a22a2a23a15a25a2a26a2a120
a121 a65 a96 a78a2a109a2a41a2a86a2a122a2a75a2a90a45a44a123a25a15a26a15a41a15a124a2a125a15a50a2a27a15a75a15a90a2a41a15a124a15a125a45a62
a71a2a73a2a48a2a49 Pa44a72a126a2a127a2a128a2a129a2a83a2a118a2a130a2a122a15a131a2a26a104a132
(1)P(A + B) = P(A) + P(B) ;
(2)P(Ac) = 1 P(A);
(3)A B =) P(A) 6 P(B);
2,a4a2a5a2a6a2a7
a133a2a134a2a135 a41 a96 a78a2a74a32a105a34a27a2a136a2a137 a70a2a138 a41a15a75a2a90a45a62a72a46a15a71a2a139a15a97a2a122a15a77a2a78a15a25a2a26a45a44 a133a15a134a2a135 a41
a140a2a141a2a50a2a27a2a142a2a117a2a57a2a58a2a41a2a48a15a49a15a143a2a144a15a145a45a62a123a73a15a27a45a44a123a116a15a117a15a146a2a147a15a148a2a149a15a150a15a48a2a151a15a41a2a52a15a152a15a153a2a154a104a132
(1) = f!1; !2; ; !ng;
(2)P(!k) = 1=n; k = 1; 2; ; n;
a149a2a150a2a48a2a151a2a27a2a155a2a156 a96 a78a2a74a32a105a102a41 a133a15a157a15a158 a41a15a48a2a49a15a159a15a151a45a62
a160 1.1 a93a2a161a2a97a2a162a2a163a2a164a2a41a2a165a2a166a76a44a123a25a15a26 A = f
a167a34a168a2a169
a69 ga44a72a170 P(A)a62
a171 a127a15a29a15a30a15a41a15a42a15a103 a70a173a172 a122a32a132
a167a174a168a175a169
a69a175a176
a167a174a168a175a169
a118a177a44a178a142a175a117a175a27a175a145a173a33a175a35a175a41a177a62a178a73
a27a76a44
P(A) = 1=2:
a160 1.2
a179
a30 100a26a2a180a2a181a45a44a107a182a32a31 a70 5a26a2a27a2a98a2a181a45a62a107a183a2a184 50a26a45a44a107a170a2a182a32a31a34a185 a70
1
Ch1 2
a172 a26a2a27a2a98a2a181a2a41a2a48a2a49a76a62
a171a187a186 a31a174a184a188a36 50 a26a175a189 a70 C50
100 a190a175a191a175a192
a44a102a182a188a31a174a185 a70 2 a26a175a98a175a181a175a41a175a184
a192
a189 a70
C4895C25
a190
a44a72a73a2a27a76a44a72a184a32a36 50a26a2a185 a70a2a172 a26a2a27a2a98a2a181a2a41a2a48a2a49a15a65
C50100
C4895C25 =
8
25:
a160 1.3
a193
a58a195a194a34a196a34a197a188a198a79a97a15a122a15a199 a69a15a70 n a122a2a152a2a58a45a44 A = f
a200a2a201
a70a2a172 a122a2a202a2a58a203a194a67a144
a204 ga44a72a170a2a25a2a26 Aa57a2a58a2a41a2a48a2a49a76a62
a171a206a205a2a207 a170 Aa57a2a58a2a41a2a48a2a49a2a208a2a40a2a209a15a210a104a44a63a108a15a83a15a116a15a117a15a211a2a212a15a213 Aa41a2a214a2a75 Ac a57a2a58
a41a2a48a2a49a76a62
P(A) = 1 P(Ac) = 1 A
n
365
365n:
a215 n=40a44 45a44 50a44 55
a216
a44a107a83 a69 a48a2a49a2a82a15a217a15a145a15a73 0.89a44 0.94a44 0.97a44 0.99a62a107a218a2a97a2a42
a103a2a219 a205 a99a2a41a2a220a2a221a2a208a2a40a2a144a15a222a45a62
a149a2a150a2a48a2a151a32a31 a70 a97a2a223a101a196a102a197a45a44a107a50a15a27a15a224a15a225a101a196a34a197a104a62 a134a15a135 a41a15a226a104a44a63a50a15a27a15a97a15a122a15a209a15a227a32a31 a70
a a122a2a228a2a225a45a44 b a122a2a229a2a225a45a44a107a139a2a98 a186 a209a15a227a101a31a102a224a101a36a102a97a15a122a15a225a104a44a107a170 na98a32a31a34a185a2a230 a70 k a122a2a228a2a225
a41a2a48a2a49 p
k
a62
a71a2a231a76a44a72a126a2a127a2a82a2a65 a172
a190
a140a2a232a104a132 a70a15a233a101a234a102a176a2a235a15a233a101a234 a62
a70a2a233a32a234
a216
a44a236a139a2a98a175a41a15a140a15a232a175a143a15a144 a204 a44 n a98a32a31 a70 k a98a2a224a2a147a2a228a2a225a45a44 n k a98a2a224a2a147
a229a2a225a76a62a72a73a2a27 a70
pk = Ckn a
kbn k
(a + b)n = C
k
n
a
a + b
k b
a + b
n k:
a235a2a233a32a234
a216
a44 a a122a2a228a2a225a32a31a34a184a32a36 k a122a76a44 b a122a2a229a2a225a32a31a34a184a32a36 n k a122a76a44a72a73a2a27
pk = C
k
aC
n k
b
Cna+b,
a69a2a237 a41a2a91 1.2a50a2a238a2a73 a235a2a233a32a234 a41a2a224a2a225a32a196a34a197a45a62
x2 a239a2a240
a6a2a7
a149a2a150a2a48a2a151a2a241a2a51a2a145a2a48a2a49a76a44a79a89a2a242a15a41a2a243a15a125a2a148 a96 a78a15a74a32a105a102a65 a70a15a138 a41a15a97a2a87a101a196a34a197a45a62a72a244a15a27
a215a15a96 a78a15a74a101a105a102a65 a235a175a138
a216
a44a245a149a175a150a175a48a175a151a175a50a175a40a175a35a175a246a175a247a175a148a177a62a245a73a175a27a177a44a245a116a175a117a175a248a175a249a175a130a175a250a175a48a175a151a177a44
a251a2a65a2a149a2a150a2a48a2a151a2a28 a235a2a138 a140a15a141a15a41a2a252a15a253a45a62
2
Ch1 3
a211a2a254a2a118 a237 a41a2a97a2a122a2a91a2a95a76a62
a160 2.1
a193a255
a1 a196a38a197a188a198 a172 a202a2a144
a255
a86a3a2 5a109
a200 6a109a24a28a24a86a5a4 a1a15a237 a44a72a211a2a147a7a6a15a145a7a8
a9 a97a2a202 20a82a7a10a76a44a12a11
a216
a50a2a155a7a13a45a62a123a170a15a218 a172 a122a15a202a15a35 a1a15a237 a41a15a48a15a49a45a62
a171 a83 xa44 ya82a24a217a24a84a24a85 a172 a202a24a147a5a14a24a41
a216
a55a76a44a16a15 a1a24a237 a41a7a17a5a18a2a131a2a26a24a27 jx yj 6 20a62
a64
G = f(x; y)j0 6 x 6 60; 0 6 y 6 60g;
g = f(x; y)jjx yj 6 20; (x; y) 2 Gg:
a19 a158a21a20 1-1a44a72a33a7a22 a172 a202a2a35 a1a2a237 a41a2a48a2a49a15a65
P(A) = S(g)S(G) = 60
2 402
602 =
5
9:
y
60
20
O
g
20 60 x
a20 1.1
a28a15a218a3a23a45a44 a116a15a117a3a24a3a25 a69a27a26 a120a15a148 a172 a202a15a147a27a14a175a41
a216
a105a174a28 5 a109
a200 6
a109a3a28a101a105a102a41a15a23 a1
a163a2a145a45a44 a46a7a29a7a30a15a40a3a31a3a32a104a62 a71a175a73a15a97a3a33a15a41a15a140a15a141a104a44 a116a175a117a15a143a15a33a15a83a15a51 a20 a141a175a84a15a85a101a36 a44a35a34a15a110a15a111
a29a24a30a24a42a24a103a24a41 a20 a141a2a84a2a85a2a65 Ga44a36a29a24a183a24a250 a172 a109a24a41a24a23 a1 a163a2a145a45a44 a46 a70 a241a2a73a2a25a2a26 Aa41a24a29a24a30
a42a2a103a2a41 a20 a141a2a84a2a85a2a65 ga44a12a15a2a25a2a26 Aa57a2a58a2a41a2a48a2a49a2a65
P(A) = S(g)S(G):
a218a7a23a2a41 S( )a33a2a83a2a84a2a85 a20 a141a2a41 a237a7a37 a44a123a39a15a33a15a83a15a84a2a85a3a38a7a39a45a44a63a111 a37 a145a45a62
a118 a237 a80a7a40a32a36a34a97a2a122a2a91a2a95a76a62
a160 2.2
a193 Buffon a196a34a197a188a198a42a41 a237a2a69 a56a3a43a15a97a15a87a3a41a3a44a3a45a104a44a107a142a15a117a3a28a101a105a102a41a3a46a2a155a15a143a15a145
a73 aa44a48a47a34a231a7a41 a237a2a69 a183a3a49a15a41a15a93a15a161a15a97a3a38a3a39a15a65 l(l 6 a) a41a7a50a45a44a107a170a2a231a7a50a2a219a15a86a15a97a3a41a3a44a3a45
a144a7a51a2a41a2a48a2a49a76a62
3
Ch1 4
a
’ x
l
x
a
2
O
g
l
2 sin ’
’
a20 1.2
a171 a83 x a84a2a85a7a50a2a41a32a31a102a109a15a147 a133a3a52 a41a175a97a15a131a3a41a27a44a3a45a175a41a3a46a15a155a104a44 ’ a84a2a85a2a127a7a41a7a44a3a45a15a219
a50a2a41a7a53a7a54
a193
a158a21a20 1-2a198a44a72a113a2a115 a70
0 6 x 6 a2; 0 6 ’ 6 ;
a55 a231a76a44
G = f(’; x)j0 6 x 6 a2; 0 6 ’ 6 g;
a46 a215 a50a2a219a7a41a7a44a7a45a2a144a7a51
a216
a44 a70 x 6 l
2 sin ’
a44a72a108a2a83 a70
g = f(’; x)jx 6 l2 sin ’; (’; x) 2 Gg:
a218 a96 a50a2a33a2a83a2a146a2a147
P(A) = S(g)S(G) =
R
0
l
2 sin ’d’
a
2
= 2 la,
a28a2a254 a207 a118a2a54a2a41a2a218a2a122a32a196a34a197a76a62
a160 3.3
a193 Berfrand a56a7a57
a198a112a28a7a58a7a59a2a65 1a41a21a60a7a61a34a22a2a23a2a41a15a184a15a97a15a131a3a62a104a44a178a196a34a182a3a38
a39a7a63a7a11a2a127a21a60a7a61 a207 a145a7a64a7a65a3a54a15a141a2a41a3a64a7a38 p3a41a2a48a2a49a2a27a7a66
a201a68a67
a218a15a27a15a97a15a122a15a130a15a250a15a48a175a151a175a41a188a196a174a197a177a44a245a244a175a27a175a71a70a69 a22a175a23a27a4a21a71a245a97a3a72a175a41a27a73 a121 a41a173a40 a204 a246a27a74a177a44
a33a2a83a2a146a2a147a2a40 a204 a41a7a75a7a76a45a62a123a118 a237 a40a101a36a102a182a32a31a102a41a7a65
a190
a62
a171 1 a183a2a250a7a62a2a143 a176 a60a78a77a27a51a15a73 a172 a109a104a62a236a40a3a79a15a97a27a33a15a59a104a44a236a211a68a80a102a120a15a182a101a31a102a97a175a109a15a28a68a60a78a77
a69 a44a63a83a2a231a2a109a2a65a2a97a7a81a15a109a15a251a68a60a102a41a68a61 a207 a145a3a64a3a65a3a54a15a141a104a44a63a119a68a60a78a77a15a82a15a65a3a65a7a82a15a145a3a38a15a41a3a83a104a62a63a113
a115a76a44a85a84 a70a2a215 a62a2a41 a9 a97a7a81a15a109a7a86a15a28a2a219a15a127a7a81a15a109a2a40a15a144a7a87a15a41a2a97a3a82a7a83 a69 a41
a216
a8a45a44a12a62a3a38a7a88 a1
a63a7a11 p3a62a12a89a2a108a2a170a2a41a2a48a2a49a2a65 1
3
a62 a158a21a20 1-3a62
4
Ch1 5
a171 2 a62a7a38a7a84a7a90a2a142a2a41a3a62a27a91a3a46 a70a3a92 a44a236a219
a191
a47 a235a3a92 a44a236a73a15a27a15a33a15a83 a26a3a93 a142a3a94 a205 a73
a86a24a97a24a131 a205 a59a76a62 a71a24a73a2a218 a96 a41a5a62a45a44 a182a7a38a7a39a24a60a2a73 p3a215 a29a5a95 a215 a142a24a41a5a62a5a91a7a46a2a61a2a73 1
2
a62 a55
a231a2a108a2a170a2a41a2a48a2a49a2a65 1
2
a62
a171 3 a62a7a96a2a182a32a31a34a109a7a97a2a97a2a247a2a120a45a44a98a89 a215 a29a7a95 a215 a182a32a31a102a109a15a238a2a73a3a58a3a59a2a65 1
2
a41 a204 a91a21a60
a216
a44a12a62a7a38a7a88a2a60a2a73 p3a62a72a46a2a231a2a61a21a60 a237a7a37 a65a2a60a68a60 a237a3a37 a41 1
4
a44a12a89a2a108a2a170a2a41a2a48a2a49a2a65 1
4
a62
(a) (b) (c)
d
a20 1.3
a204 a97a32a196a34a197a2a146a2a147a7a65
a190
a40 a204 a41a3a75a3a76a104a44a100a99 a55 a27a3a101a15a51a15a148a15a40 a204 a41a15a145a2a33a15a35a15a59a104a62a107a28a15a246 1
a31 a44 a26 a120a24a97a7a102a21a80a102a120a2a46 a9 a97a7a102a15a28a21a60a103a77 a69 a163a15a164a2a82a7a104a106a105a112a246 2 a31 a26a5a93 a62a24a41a32a31a34a109a2a28a15a97a21a80
a120 a205 a59 a69 a163a2a164a2a82a7a104a107a105a123a46a15a246 3 a15 a26 a120a7a62a2a41a32a31a34a109a2a28a21a60a3a61a34a163a15a164a15a82a3a104a45a62 a55 a231a45a44a108a65
a190
a40
a204 a41a3a75a3a76a15a27a3a50a15a71a3a65
a190
a40 a204 a41a15a22a175a23a175a29a15a30
a193a110a109a27a111
a254a3a112a175a54a175a144 a204 a198a44 a71a175a73a3a113a115a114 a41a15a22a175a23
a29a2a30a2a46a7a116a76a44a72a142a2a117a2a143a15a27a3a117a7a118a15a41a45a62
a55 a231a2a28a7a119a2a51a120a69 a22a2a23a107a71a122a121a123a69 a183a15a184a107a71a122a121a123a69 a145a15a33a2a35a106a71a124a121a123a69 a163a15a164a106a71
a216
a44a123a126a15a127a21a125a78a118a15a182a7a73 a121 a62
x3 a6a2a9a7a126a7a127a2a8a2a6a2a9a2a10a2a11a2a13a15a14a2a16
a128 a172a5a129 a40a177a36a38a148a24a48a2a49 a205 a99 a69 a41a24a120 a121 a44 a207 a118a2a54a2a119a7a130a7a131a7a118a2a41a2a153a2a154a2a48a2a49a7a132a7a39 a176 a48
a49a2a74a32a105 a62
a128 a237a21a133a103a134a7a135 a11a76a44a79a25a2a26a2a50a2a27a2a86a15a87 a96 a78a2a109a15a88a2a89a15a41a2a75a15a90a45a44a72a25a2a26a3a28a32a105a102a41a2a124a15a125a2a39a15a50
a27a24a75a24a90a24a124a24a125a45a62a72a244a2a27a104a44a79a208a7a136 a70 a71a15a75a2a90a15a249a7a44 a138a7a137 a62a72a71a2a73a15a25a2a26a45a44a72a97a2a122a3a138a21a125a102a113a2a41a3a18a2a170a45a44
a50a24a27a24a108 a70 a25a2a26a15a88a2a89a2a41a15a75a2a90a2a71a15a73a2a208
a193 A [Ba198a139a121a42a51 a193 A \B a140
a64a24a65 ABa198a139a121a42a141
a193 Aca198
a145a5a65
a190
a124a24a125a5a142a21a143a106a62a144a128a7a145a7a146a2a152a3a6a7a147a7a148a7a149a7a150a7a151a7a152a2a73 1933
a153
a28a155a154 a48a24a49
a57
a77a7a156a2a48a7a157a107a158
a97a7a159a32a31 a44a79a51a7a160a7a161a7a162a2a41
a191a15a192
a219a15a75a2a90
a57
a41a15a99a2a109a15a89a2a242a3a4a2a246a15a247a2a148a15a218a2a122a101a196a34a197a45a44a12a163a101a36a34a148a15a48
a49a2a74a32a105a34a41a2a48a7a157a76a62
1 a6a2a9a7a126a7a127a7a164a7a165a7a166a7a167a7a168
(1)a96 a78a2a74a32a105
5
Ch1 6
a27a2a97a7a169a2a74a2a75a2a90a76a44a72a65 a96 a78a2a74a101a105a106a62
(2) F a219a2a33a7a132a2a74a32a105
F a27a21a170 a41a2a86a2a87a2a95a2a75a2a108a7a171a2a89a2a41a2a97a2a122 a172
a52a76a44a12a173a2a142a2a128a2a129a45a132
(1) 2 F ;
(2) a34 A 2 Fa44a12a15 Ac 2 F;
(3) a34 An 2 Fa44 n =1a44 2a44 a44a12a15
1[
n=1
An 2 F;
a174 ( a44 F)a65a2a33a7a132a2a74a32a105 a62
(3) P a219a2a48a2a49a2a74a32a105
a48a2a49 P a27a2a120 a121 a28 F a69 a41a7a175a2a52a76a44a12a173a2a142a2a27a15a97a2a122 a186 F a147 [0; 1]a41a7a176a7a177a45a132
P,F ! [0; 1]a44a12a29a2a142a2a128a2a129a45a132
(1) 8A 2 Fa44 P(A) > 0 ;
(2) P( ) = 1;
(3) a34 An 2 Fa44 n =1a44 2a44 a44a12a29 An a30a2a40a2a144a7a51a76a44a12a15
P(
1[
n=1
An) =
1X
n=1
P(An);
a174 a218 a96 a41 Pa65 a33a178a132 a74a104a105 ( ;F)a69 a41 a97 a122 a48 a49a178a132a178a39
a193
a134 a174a24a65a24a48 a49a101a198a44a179a174 ( ;F; P)
a65a2a97a2a48a2a49a2a74a32a105 a62
a33a2a83a2a252a32a36 a44a72a48a2a49a7a132a7a39 P a70 a83a2a118a2a59a7a180a45a132
(1) P(;) = 0 ;
(2) a34 Ak 2 Fa44 k =1a44 2a44 a44 na44a12a29 Ak a30a2a40a2a144a7a51a76a44a12a15
P(
n[
k=1
Ak) =
nX
k=1
P(Ak);
(3) a34 P(Ac) = 1 P(A);
(4) P(A B) = P(A) P(B) ;
(5) a34 A 2 Ba44a12a15 P(A) 6 P(B) ;
(6) P(A[ B) 6 P(A) + P(B) ;
(7) P(Snk=1 Ak) 6 Pnk=1 P(Ak) ;
6
Ch1 7
(8) a34 A1 2 A2 2 a44a12a15
P(
1[
n=1
) = lim
n!1
P(An);
2 a181a7a182
a13a7a183
a170 a83 a69 a48 a49 a41 a59a178a180 a33a24a83 a146a24a147a5a184
a192
a160a5a185a24a41 a134a24a135 a140a24a141a76a132 a71 a183a5a49 a172 a122a24a25a24a26 Aa44 Ba44
a70
P(A[ B) = P(A) + P(B) P(AB):
a134 a18a7a186a21a125a34a92a15a118a177a132a187a170 A [ B = A [ BAca44 B = BA [ BAca44a35a29 A a219 BAc a40a2a144
a51a76a44 BA a219 BAc a40a2a144a7a51a76a44 a70
P(A [ B) = P(A) + P(BAc);
P(BAc) = P(B) P(BA);
a172 a185a2a144a7a184a7a173a2a146a2a108a7a18a2a41a7a160a3a185a45a62
a71a2a73a7a66a2a122a2a75a2a90a2a41a2a140a2a141a76a44a123a116a15a117 a70 a118 a237 a41 Jordan a160a7a185a45a132
P(
n[
i=1
Ai) =
nX
k=1
( 1)k+1
X
16i1<i2< <ik6n
P(Ai1Ai2 Aik):
a51a2a209a7a188a7a99a7a161
a140
a6a2a52a2a152a7a189a3a190
a192
a163a15a33a7a40a101a36a78a186a21a125a106a62
a118 a237 a40a32a36a34a97a2a122a2a91a2a95a76a62
a160 3.1
a193a192a191
a71a32a196a34a197a188a198 n a71a7a152a7a193a2a22a2a23a7a194a7a195a7a196a15a28a3a38a3a197a15a41 a172a3a198 a44a107a211a15a58a15a110a7a199a3a196a15a28
a182a32a31a34a41a2a97 a198 a44 a196
a200a15a201
a70 a97a15a71a7a152a3a193 a237 a71 a237 a196a15a41a15a48a2a49a15a27a3a66
a201a155a67
a171 a64
B =a200a2a201
a70 a97a2a71a7a152a7a193 a237 a71 a237 a196a45a44
Ai= a200 ia71a7a152a7a193 a237 a71 a237 a196a76a44 i =1a44 2a44a103a201 a44 na62
a210a7a22 B = Sn
i=1 Ai
a62a72a71a2a183a7a49a2a41 1 6 i
1 < i2 < < ik 6 n
a44 a70
P(Ai1Ai2) Aik = (n k)!n! ;
X
16i1<i2< <ik6n
P(Ai1Ai2 Aik) = Ckn (n k)!n! = 1k!;
7
Ch1 8
a73a2a27a76a44a202a170 Jordan a160a7a185a2a33a7a22
P(B) =
nX
k=1
( 1)k+1 1k!:
a215 n ! 1
a216
a44 P(B) =
nX
k=1
( 1)k+1 1k! ! 1 e 1a62
8
