Руденко В.М. Математична статистика
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81
Ɋɿɲɟɧɧɹ: Ʉɿɥɶɤɿɫɬɶ ɩɨɹɜɥɟɧɶ ɛɚɠɚɧɢɯ ɩɨɞɿɣ m=1, ɡɚɝɚɥɶɧɚ ɤɿɥɶɤɿɫɬɶ ɜɢɩɚ-
ɞɤɿɜ
n=20. Ɂɧɚɱɟɧɧɹ ɣɦɨɜɿɪɧɨɫɬɿ Ɋ(Ⱥ) ɩɨɞɿʀ Ⱥ – ɰɟ ɜɿɞɧɨɲɟɧɧɹ m / n:
%505,0
20
1
)(
n
m
AP
.
ȼɿɞɩɨɜɿɞɶ: ɣɦɨɜɿɪɧɿɫɬɶ ɜɢɬɹɝɬɢ ɡ ɩɟɪɲɨɝɨ ɪɚɡɭ ɛɿɥɟɬ ʋ7 ɫɤɥɚɞɚɽ 0,05 ɚɛɨ 5%.
ɉɪɢɤɥɚɞ 3.3. ɋɬɭɞɟɧɬ ɡɧɚɽ ɜɿɞɩɨɜɿɞɶ ɥɢɲɟ ɧɚ 5 ɟɤɡɚɦɟɧɚɰɿɣɧɢɯ ɛɿɥɟɬɿɜ ɿ ɧɟ
ɡɧɚɽ ɜɿɞɩɨɜɿɞɿ ɧɚ ɪɟɲɬɭ 15 ɛɿɥɟɬɿɜ. əɤɚ ɣɦɨɜɿɪɧɿɫɬɶ ɬɨɝɨ, ɳɨ ɩɟɪɲɢɣ ɜɢɬɹɝɧɭ-
ɬɢɣ ɧɚɜɦɚɧɧɹ ɛɿɥɟɬ ɜɢɹɜɢɬɶɫɹ ɬɚɤɢɦ, ɧɚ ɹɤɢɣ ɫɬɭɞɟɧɬ ɡɧɚɽ ɜɿɞɩɨɜɿɞɿ?
Ɋɿɲɟɧɧɹ: Ɂɚɝɚɥɶɧɚ ɤɿɥɶɤɿɫɬɶ ɛɿɥɟɬɿɜ ɫɤɥɚɞɚɽ 5+15=20 (n=20), ɫɩɪɢɹɬɥɢɜɢɯ
ɞɥɹ ɫɬɭɞɟɧɬɚ ɪɟɡɭɥɶɬɚɬɿɜ ɜɫɶɨɝɨ 5 (
m=5). Ɂɜɿɞɫɢ ɣɦɨɜɿɪɧɿɫɬɶ ɛɚɠɚɧɨʀ ɩɨɞɿʀ:
%2525,0
20
5
)(
n
m
AP
ȼɿɞɩɨɜɿɞɶ: ɣɦɨɜɿɪɧɿɫɬɶ ɜɢɬɹɝɬɢ ɛɚɠɚɧɢɣ ɛɿɥɟɬ ɫɤɥɚɞɚɽ 0,25 ɚɛɨ 25%.
Ɉɬɠɟ, ɣɦɨɜɿɪɧɿɫɬɶ ɩɨɞɿʀ ɽ ɨɫɧɨɜɧɢɦ ɩɨɧɹɬɬɹɦ ɬɟɨɪɿʀ ɣɦɨɜɿɪɧɨɫɬɿ, ɩɪɨɬɟ
ɪɨɡɝɥɹɧɭɬɿ ɤɥɚɫɢɱɧɿ ɨɡɧɚɱɟɧɧɹ ɣɦɨɜɿɪɧɨɫɬɟɣ, ɚ ɬɚɤɨɠ ɧɚɜɟɞɟɧɿ ɩɪɢɤɥɚɞɢ ɞɚ-
ɸɬɶ ɥɢɲɟ ɡɚɝɚɥɶɧɟ ɿɧɬɭʀɬɢɜɧɟ ɭɹɜɥɟɧɧɹ ɳɨɞɨ ɨɰɿɧɤɢ ɬɚ ɩɪɨɝɧɨɡɭɜɚɧɧɹ ɣɦɨɜɿ-
ɪɧɨɫɬɿ. ɐɿ ɦɟɬɨɞɨɥɨɝɿɱɧɿ ɩɿɞɯɨɞɢ ɧɟ ɞɚɸɬɶ ɫɬɪɨɝɢɯ ɱɢɫɟɥɶɧɢɯ ɡɧɚɱɟɧɶ. ɇɟ ɜɫɿ
ɩɨɞɿʀ ɦɨɠɧɚ ɜɜɚɠɚɬɢ ɪɿɜɧɨɦɨɠɥɢɜɢɦɢ, ɧɟ
ɜɫɿ ɣɦɨɜɿɪɧɨɫɬɿ ɦɨɠɧɚ ɨɰɿɧɸɜɚɬɢ ɹɤ
ɡɛɿɠɧɨɫɬɿ ɱɚɫɬɨɬ, ɧɟɹɫɧɨ ɿ ɬɟ, ɫɤɿɥɶɤɢ ɜɢɩɪɨɛɨɜɭɜɚɧɶ ɬɪɟɛɚ ɡɞɿɣɫɧɸɜɚɬɢ ɬɚ ɿɧ.
Ɋɨɡɝɥɹɧɟɦɨ ɨɡɧɚɱɟɧɧɹ ɣɦɨɜɿɪɧɨɫɬɿ ɭ ɪɚɦɤɚɯ
ɚɤɫɿɨɦɚɬɢɱɧɨɝɨ ɩɿɞɯɨɞɭ ɞɨ
ɦɚɬɟɦɚɬɢɱɧɨʀ ɦɨɞɟɥɿ, ɳɨ ɛɭɥɚ ɡɚɩɪɨɩɨɧɨɜɚɧɚ Ⱥ.Ɇ. Ʉɨɥɦɨɝɨɪɨɜɢɦ.
Ɉɡɧɚɱɟɧɧɹ.
Ƀɦɨɜɿɪɧɿɫɬɶ. ɇɟɯɚɣ ɫɤɿɧɱɟɧɚ ɦɧɨɠɢɧɚ :={Ȧ} ɽ ɩɪɨɫɬɨɪɨɦ
ɟɥɟɦɟɧɬɚɪɧɢɯ ɩɨɞɿɣ
Ȧ, ɳɨ ɜɿɞɩɨɜɿɞɚɸɬɶ ɞɟɹɤɨɦɭ ɫɬɨɯɚɫɬɢɱɧɨɦɭ
9
ɞɨɫɥɿɞɨɜɿ.
ɇɟɯɚɣ ɤɨɠɧɿɣ ɟɥɟɦɟɧɬɚɪɧɿɣ ɩɨɞɿʀ
Ȧ, ɹɤɚ ɧɚɥɟɠɢɬɶ ɞɨ ɦɧɨɠɢɧɿ :, ɬɨɛɬɨ
:
Z
, ɩɨɫɬɚɜɥɟɧɨ ɭ ɜɿɞɩɨɜɿɞɧɿɫɬɶ ɧɟɜɿɞ’ɽɦɧɟ ɱɢɫɥɨ P(Ȧ), ɬɨɛɬɨ P(Ȧ)0. ɑɢɫ-
ɥɨ
P(Ȧ) ɨɡɧɚɱɢɦɨ ɹɤ ɿɦɨɜɿɪɧɿɫɬɶ ɟɥɟɦɟɧɬɚɪɧɨʀ ɩɨɞɿʀ Ȧ, ɩɪɢɱɨɦɭ ɫɭɦɚ ɣɦɨɜɿ-
ɪɧɨɫɬɟɣ ɜɫɿɯ ɟɥɟɦɟɧɬɚɪɧɢɯ ɩɨɞɿɣ ɞɨɪɿɜɧɸɽ 1, ɬɨɛɬɨ:
1)(
¦
:
Z
Z
P . (3.3)
ɉɚɪɚ {
:, Ɋ} ɽ ɿɦɨɜɿɪɧɿɫɧɢɦ ɩɪɨɫɬɨɪɨɦ, ɹɤɢɣ ɫɤɥɚɞɚɽɬɶɫɹ ɡɿ ɫɤɿɧɱɟɧɨʀ
9
ɋɬɨɯɚɫɬɢɱɧɢɣ (ɜɿɞ ɝɪɟɰ. stochastikos – ɫɩɪɨɦɨɠɧɢɣ ɭɝɚɞɭɜɚɬɢ), ɜɢɩɚɞɤɨɜɢɣ, ɿɦɨɜɿɪɧɿɫɧɢɣ.
82
ɦɧɨɠɢɧɢ : ɿ ɧɟɜɿɞ’ɽɦɧɨʀ ɮɭɧɤɰɿʀ Ɋ, ɹɤɚ ɜɢɡɧɚɱɟɧɚ ɧɚ ɦɧɨɠɢɧɿ : ɿ ɡɚɞɨɜɨɥɶ-
ɧɹɽ ɭɦɨɜɿ (3.3). Ɂɜɿɞɫɢ
ɣɦɨɜɿɪɧɿɫɬɶ Ɋ(Ⱥ) ɞɟɹɤɨʀ ɩɨɞɿʀ Ⱥ ɞɨɪɿɜɧɸɽ ɫɭɦɿ ɣɦɨɜɿɪ-
ɧɨɫɬɟɣ ɟɥɟɦɟɧɬɚɪɧɢɯ ɩɨɞɿɣ
Ȧ, ɳɨ ɜɯɨɞɹɬɶ ɞɨ ɩɨɞɿʀ Ⱥ:
¦
A
PAP
Z
Z
)()(
. (3.4)
Ɍɨɞɿ ɛɭɞɶ-ɹɤɭ ɱɢɫɥɨɜɭ ɮɭɧɤɰɿɸ
Ɋ(Ⱥ), ɜɢɡɧɚɱɟɧɭ ɧɚ ɫɤɿɧɱɟɧɿɣ ɦɧɨɠɢɧɿ
:={Ȧ}, ɹɤɚ ɽ ɩɪɨɫɬɨɪɨɦ ɟɥɟɦɟɧɬɚɪɧɢɯ ɩɨɞɿɣ Ȧ, ɧɚɡɢɜɚɸɬɶ ɿɦɨɜɿɪɧɿɫɬɸ, ɹɤɳɨ
ɜɢɤɨɧɭɸɬɶɫɹ ɬɪɢ ɭɦɨɜɢ (ɚɤɫɿɨɦɢ Ʉɨɥɦɨɝɨɪɨɜɚ):
1)
Ɋ(Ⱥ) 0 ɞɥɹ ɛɭɞɶ-ɹɤɨʀ :Ⱥ ;
2)
Ɋ(:)=1;
3)
Ɋ(Ⱥ
1
*
Ⱥ
2
*
Ⱥ
3
*
...) = Ɋ(Ⱥ
1
)+ Ɋ(Ⱥ
2
)+ Ɋ(Ⱥ
3
)+... ɞɥɹ ɩɨɩɚɪɧɨ ɧɟɫɭɦɿɫ-
ɧɢɯ ɜɢɩɚɞɤɨɜɢɯ ɩɨɞɿɣ (
Ⱥ
i
Ⱥ
j
= Ø, i j).
Ɉɬɠɟ, ɫɤɨɧɫɬɪɭɣɨɜɚɧɨ ɦɚɬɟɦɚɬɢɱɧɢɣ ɨɛ’ɽɤɬ, ɹɤɢɣ ɦɨɠɧɚ ɡɚɫɬɨɫɨɜɭɜɚɬɢ
ɩɪɢ ɩɨɛɭɞɨɜɿ
ɿɦɨɜɿɪɧɿɫɧɢɯ ɦɨɞɟɥɟɣ. ɇɚɩɪɢɤɥɚɞ, ɜɢɩɪɨɛɭɜɚɧɧɹɦ ɡ ɩɿɞɤɢɞɚɧ-
ɧɹɦ ɦɨɧɟɬɢ ɜɿɞɩɨɜɿɞɚɽ
ɿɦɨɜɿɪɧɿɫɧɢɣ ɩɪɨɫɬɿɪ {:, Ɋ}, ɞɟ : = {Ƚ,ɐ} – ɦɧɨɠɢɧɚ
ɟɥɟɦɟɧɬɚɪɧɢɯ ɩɨɞɿɣ;
P(Ƚ) = P(ɐ) = ½ – ɣɦɨɜɿɪɧɨɫɬɿ ɟɥɟɦɟɧɬɚɪɧɢɯ ɩɨɞɿɣ; ɩɨ-
ɡɧɚɱɟɧɧɹ ɟɥɟɦɟɧɬɚɪɧɢɯ ɩɨɞɿɣ:
Ƚ – «ɜɢɩɚɜ ɝɟɪɛ», ɐ – «ɜɢɩɚɥɚ ɰɢɮɪɚ».
Ⱥɤɫɿɨɦɚɬɢɱɧɟ ɨɡɧɚɱɟɧɧɹ ɣɦɨɜɿɪɧɨɫɬɿ
Ɋ(Ⱥ) ɩɨɝɨɞɢɬɶɫɹ ɡ ɿɧɬɭʀɬɢɜɧɢɦ, ɡɝɿɞɧɨ ɡ
ɹɤɢɦ ɣɦɨɜɿɪɧɿɫɬɶ ɩɨɞɿʀ
Ⱥ – ɰɟ ɱɢɫɥɨ ɜɿɞ 0 ɞɨ 1, ɳɨ ɽ ɡɛɿɝɨɦ ɱɚɫɬɨɬɢ ɪɟɚɥɿɡɚɰɿʀ
ɩɨɞɿʀ
Ⱥ ɩɪɢ ɧɟɨɛɦɟɠɟɧɨɦɭ ɱɢɫɥɿ ɩɨɜɬɨɪɟɧɶ ɿ ɩɨɫɬɿɣɧɢɯ ɭɦɨɜɚɯ ɜɢɩɪɨɛɭɜɚɧɶ.
Ɂ ɜɢɡɧɚɱɟɧɧɹ ɿɦɨɜɿɪɧɨɫɬɿ ɩɨɞɿʀ, ɚ ɬɚɤɨɠ ɭɦɨɜ (3.3) ɿ (3.4) ɜɢɩɥɢɜɚɸɬɶ ɿɧɲɿ
ɜɥɚɫɬɢɜɨɫɬɿ ɣɦɨɜɿɪɧɨɫɬɟɣ:
4)
Ⱦɥɹ ɛɭɞɶ-ɹɤɨʀ ɩɨɞɿʀ Ⱥ ɣɦɨɜɿɪɧɿɫɬɶ ɩɪɨɬɢɥɟɠɧɨʀ ɩɨɞɿʀ
)(1)( APAP
.
5)
ɋɭɦɚ ɣɦɨɜɿɪɧɨɫɬɟɣ ɩɪɨɬɢɥɟɠɧɢɯ ɩɨɞɿɣ ɞɨɪɿɜɧɸɽ ɨɞɢɧɢɰɿ:
1)()( APAP
.
6)
Ƀɦɨɜɿɪɧɿɫɬɶ ɞɨɫɬɨɜɿɪɧɨʀ ɩɨɞɿʀ Ɋ(:) = 1 (ɡɚ ɚɤɫɿɨɦɨɸ 2).
7)
Ƀɦɨɜɿɪɧɿɫɬɶ ɧɟɦɨɠɥɢɜɨʀ ɩɨɞɿʀ Ɋ(
:
) = 1 – Ɋ(:) = 1 – 1 = 0.
8)
Ƀɦɨɜɿɪɧɿɫɬɶ ɞɨɛɭɬɤɭ Ⱥ·ȼ ɫɭɦɿɫɧɢɯ ɩɨɞɿɣ Ⱥ ɿ ȼ:
)()()( B
P
A
P
ȼȺ
P
. Ɂ ɞɿ-
ɚɝɪɚɦɢ ɪɢɫ. 3.2ɚ ɜɢɞɧɨ, ɳɨ ɫɭɦɿɫɧɿ ɩɨɞɿʀ
Ⱥ ɿ B ɦɚɸɬɶ ɡɚɝɚɥɶɧɭ (ɫɭɦɿɫɧɭ) ɩɥɨɳɭ
83
ɩɨɞɿɣ (ɡɚɮɚɪɛɨɜɚɧɚ ɩɥɨɳɚ), ɬɨɦɭ ɣɦɨɜɿɪɧɿɫɬɶ ɞɨɛɭɬɤɭ ɫɭɦɿɫɧɢɯ ɩɨɞɿɣ Ɋ(Ⱥ·ȼ )> 0.
9)
Ƀɦɨɜɿɪɧɿɫɬɶ ɫɭɦɢ Ⱥ+ȼ ɫɭɦɿɫɧɢɯ ɩɨɞɿɣ Ⱥ ɿ ȼ ɜɢɡɧɚɱɚɽɬɶɫɹ ɮɨɪɦɭɥɨɸ:
)()()()( BA
P
B
P
A
P
ȼȺ
P
, ɞɟ
)( BA
P
– ɣɦɨɜɿɪɧɿɫɬɶ ɞɨɛɭɬɤɭ ɩɨɞɿɣ Ⱥ ɿ
ȼ. Ɂ ɞɿɚɝɪɚɦɢ ɪɢɫ. 3.2ɛ ɜɢɞɧɨ, ɳɨ ɩɨɞɿʀ Ⱥ ɿ ȼ ɦɚɸɬɶ ɫɭɦɿɫɧɭ ɩɥɨɳɭ ɩɨɞɿɣ, ɹɤɚ
ɦɟɧɲɚ ɡɚ ɫɭɦɭ ɨɤɪɟɦɨ ɜɡɹɬɢɯ ɩɨɞɿɣ
Ⱥ ɿ ȼ ɧɚ ɜɟɥɢɱɢɧɭ ɞɨɛɭɬɤɭ Ⱥ·ȼ ɩɨɞɿɣ Ⱥ ɿ
ȼ. ȼɿɞɩɨɜɿɞɧɨ ɣɦɨɜɿɪɧɿɫɬɶ ɫɭɦɢ ɫɭɦɿɫɧɢɯ ɩɨɞɿɣ Ɋ(B+C) < Ɋ(ȼ)+ Ɋ(ɋ ).
ɚ)
)()()( B
P
A
P
ȼȺ
P
ɛ)
)()()()(
B
A
P
B
P
A
P
ȼ
Ⱥ
P
Ɋɢɫ. 3.2. Ƀɦɨɜɿɪɧɨɫɬɿ ɞɨɛɭɬɤɭ ɿ ɫɭɦɢ ɫɭɦɿɫɧɢɯ ɩɨɞɿɣ Ⱥ ɿ B
10)
Ƀɦɨɜɿɪɧɿɫɬɶ ɞɨɛɭɬɤɭ Ⱥ·ȼ ɧɟɫɭɦɿɫɧɢɯ ɩɨɞɿɣ Ⱥ ɿ ȼ ɞɨɪɿɜɧɸɽ ɧɭɥɸ. Ɂ ɞɿɚ-
ɝɪɚɦɢ ɪɢɫ. 3.3ɚ ɜɢɞɧɨ, ɳɨ ɧɟɫɭɦɿɫɧɿ ɩɨɞɿʀ
Ⱥ ɿ ȼ ɧɟ ɦɚɸɬɶ ɫɩɿɥɶɧɨʀ ɩɥɨɳɿ, ɩɟ-
ɪɟɬɢɧ ɜɿɞɩɨɜɿɞɧɢɯ ɩɿɞɦɧɨɠɢɧ ɽ ɩɨɪɨɠɧɹ ɦɧɨɠɢɧɚ Ø ɩɨɞɿɣ, ɬɨɦɭ
ɣɦɨɜɿɪɧɿɫɬɶ
ɞɨɛɭɬɤɭ ɧɟɫɭɦɿɫɧɢɯ ɩɨɞɿɣ Ɋ(Ⱥ·ȼ) = 0.
11)
Ƀɦɨɜɿɪɧɿɫɬɶ ɫɭɦɢ Ⱥ+ȼ ɧɟɫɭɦɿɫɧɢɯ ɩɨɞɿɣ Ⱥ ɿ ȼ ɜɢɡɧɚɱɚɽɬɶɫɹ ɫɩɪɨɳɟ-
ɧɨɸ ɮɨɪɦɭɥɨɸ:
Ɋ(Ⱥ+ȼ) = Ɋ(Ⱥ) + Ɋ(ȼ). Ɂ ɞɿɚɝɪɚɦɢ ɪɢɫ. 3.3ɛ ɜɢɞɧɨ, ɳɨ ɩɨɞɿʀ Ⱥ
ɿ
ȼ ɧɟ ɦɚɸɬɶ ɫɩɿɥɶɧɨʀ ɩɥɨɳɿ ɩɨɞɿɣ, ɹɤɚ ɛ ɡɦɟɧɲɭɜɚɥɚ ɡɚɝɚɥɶɧɭ ɫɭɦɭ ɨɤɪɟɦɨ
ɜɡɹɬɢɯ ɩɨɞɿɣ
B ɿ C.
ɚ)
Ɋ(Ⱥ·ȼ) = 0 ɛ) Ɋ(Ⱥ+ȼ) = Ɋ(Ⱥ) + Ɋ(ȼ).
Ɋɢɫ. 3.3. Ƀɦɨɜɿɪɧɨɫɬɿ ɞɨɛɭɬɤɭ ɿ ɫɭɦɢ
ɧɟɫɭɦɿɫɧɢɯ ɩɨɞɿɣ Ⱥ ɿ B
12)
ɋɭɦɚ ɣɦɨɜɿɪɧɨɫɬɟɣ ɜɫɿɯ ɧɟɫɭɦɿɫɧɢɯ ɩɨɞɿɣ {Ⱥ
1
, Ⱥ
2
, …, Ⱥ
n
}, ɳɨ ɭɬɜɨ-
ɪɸɸɬɶ ɩɨɜɧɭ ɝɪɭɩɭ, ɞɨɪɿɜɧɸɽ ɨɞɢɧɢɰɿ
P(Ⱥ
1
) + P(Ⱥ
2
) +, …, + P(Ⱥ
n
) = 1 ɚɛɨ 1)(
1
¦
n
i
i
AP .
ɉɪɢ ɡɚɫɬɨɫɭɜɚɧɧɿ ɦɟɬɨɞɿɜ ɬɟɨɪɿʀ ɣɦɨɜɿɪɧɨɫɬɟɣ ɿ ɦɚɬɟɦɚɬɢɱɧɨʀ ɫɬɚɬɢɫɬɢɤɢ
84
ɜɢɤɨɪɢɫɬɨɜɭɽɬɶɫɹ ɩɨɧɹɬɬɹ ɧɟɡɚɥɟɠɧɨɫɬɿ ɩɨɞɿɣ. ɉɨɞɿʀ Ⱥ, ȼ, ɋ, ... ɽ ɧɟɡɚɥɟɠ-
ɧɢɦɢ
, ɹɤɳɨ ɣɦɨɜɿɪɧɿɫɬɶ ʀɯɧɶɨɝɨ ɫɩɿɥɶɧɨɝɨ ɡɞɿɣɫɧɟɧɧɹ ɞɨɪɿɜɧɸɽ ɞɨɛɭɬɤɨɜɿ
ɣɦɨɜɿɪɧɨɫɬɟɣ ɡɞɿɣɫɧɟɧɧɹ ɤɨɠɧɨʀ ɡ ɧɢɯ ɨɤɪɟɦɨ:
Ɋ(Ⱥ·ȼ·ɋ...) = Ɋ(Ⱥ)·Ɋ(ȼ)·Ɋ(ɋ)... .
Ɂɝɿɞɧɨ ɡ ɰɢɦ ɨɡɧɚɱɟɧɧɹɦ ɡɞɿɣɫɧɟɧɧɹ ɚɛɨ ɧɟɡɞɿɣɫɧɟɧɧɹ ɨɞɧɿɽʀ ɧɟɡɚɥɟɠɧɨʀ
ɩɨɞɿʀ ɧɟ ɩɨɜɢɧɧɟ ɜɩɥɢɜɚɬɢ ɧɚ ɡɞɿɣɫɧɟɧɧɹ ɚɛɨ ɧɟɡɞɿɣɫɧɟɧɧɹ ɿɧɲɨʀ. ɇɚɩɪɢɤɥɚɞ,
ɭ ɜɢɩɪɨɛɭɜɚɧɧɹɯ ɩɪɢ ɧɟɡɚɥɟɠɧɨɦɭ ɩɿɞɤɢɞɚɧɧɿ ɞɜɨɯ ɦɨɧɟɬ ɩɪɨɫɬɿɪ ɟɥɟɦɟɧɬɚ-
ɪɧɢɯ ɩɨɞɿɣ ɫɤɥɚɞɚɽɬɶɫɹ ɡ ɱɨɬɢɪɶɨɯ ɟɥɟɦɟɧɬɿɜ:
ȽȽ, Ƚɐ, ɐȽ, ɐɐ (ɩɨɡɧɚɱɟɧɧɹ
ɟɥɟɦɟɧɬɚɪɧɢɯ ɩɨɞɿɣ:
ȽȽ – ɞɥɹ ɩɟɪɲɨʀ ɦɨɧɟɬɢ ɜɢɩɚɜ ɝɟɪɛ ɿ ɞɥɹ ɞɪɭɝɨʀ – ɬɟɠ
ɝɟɪɛ;
ɐȽ – ɞɥɹ ɩɟɪɲɨʀ – ɰɢɮɪɚ, ɞɥɹ ɞɪɭɝɨʀ – ɝɟɪɛ ɿ ɬ.ɞ.). Ɉɫɤɿɥɶɤɢ ɩɨɞɿʀ ɬɢɩɭ
«
Ƚ – ɞɥɹ ɩɟɪɲɨʀ ɦɨɧɟɬɢ ɜɢɩɚɜ ɝɟɪɛ» ɿ «Ƚ – ɞɥɹ ɞɪɭɝɨʀ ɦɨɧɟɬɢ ɜɢɩɚɜ ɝɟɪɛ» ɽ
ɧɟɡɚɥɟɠɧɢɦɢ ɡɚ ɜɢɡɧɚɱɟɧɧɹɦ ɧɟɡɚɥɟɠɧɢɯ ɜɢɩɪɨɛɭɜɚɧɶ, ɿ ɣɦɨɜɿɪɧɿɫɬɶ ɤɨɠɧɨ-
ɝɨ ɡ ɧɢɯ ɞɨɪɿɜɧɸɽ ½, ɬɨ ɣɦɨɜɿɪɧɿɫɬɶ ɩɨɞɿʀ
ȽȽ ɞɨɪɿɜɧɸɽ ½ · ½ = ¼. Ⱥɧɚɥɨɝɿɱɧɨ
ɣɦɨɜɿɪɧɿɫɬɶ ɤɨɠɧɨɝɨ ɡ ɿɧɲɢɯ ɟɥɟɦɟɧɬɚɪɧɢɯ ɩɨɞɿɣ ɬɚɤɨɠ ɞɨɪɿɜɧɸɽ ¼. Ɂɜɿɞɫɢ
ɫɭɦɚ ɣɦɨɜɿɪɧɨɫɬɟɣ ɜɫɿɯ ɱɨɬɢɪɶɨɯ ɟɥɟɦɟɧɬɚɪɧɢɯ ɩɨɞɿɣ ɞɨɪɿɜɧɸɜɚɬɢɦɟ ɨɞɢɧɢɰɿ.
ɍɦɨɜɧɚ ɣɦɨɜɿɪɧɿɫɬɶ
əɤɳɨ ɩɨɞɿɹ Ⱥ ɜɿɞɛɭɜɚɽɬɶɫɹ ɭ ɜɢɩɪɨɛɭɜɚɧɿ, ɹɤɟ ɨɛɦɟɠɟɧɟ ɞɨɞɚɬɤɨɜɢɦɢ
ɭɦɨɜɚɦɢ ɡɞɿɣɫɧɟɧɧɹ ɩɨɞɿʀ
ȼ, ɬɨ ɦɿɪɚ ɦɨɠɥɢɜɨɫɬɿ ɩɨɞɿʀ Ⱥ ɜɢɡɧɚɱɚɽɬɶɫɹ ɭɦɨɜ-
ɧɨɸ
ɣɦɨɜɿɪɧɿɫɬɸ
)|( BA
P
. Ɉɬɠɟ, ɭɦɨɜɧɨɸ ɣɦɨɜɿɪɧɿɫɬɸ
)|( BA
P
ɧɚɡɢɜɚɽɬɶ-
ɫɹ ɣɦɨɜɿɪɧɿɫɬɶ ɩɨɞɿʀ
Ⱥ, ɨɛɱɢɫɥɟɧɚ ɡɚ ɭɦɨɜɢ, ɳɨ ɩɨɞɿɹ ȼ ɜɠɟ ɜɿɞɛɭɥɚɫɹ.
ɍɦɨɜɧɚ ɣɦɨɜɿɪɧɿɫɬɶ ɦɚɽ ɫɟɧɫ ɞɥɹ
ɡɚɥɟɠɧɢɯ ɩɨɞɿɣ. Ⱦɥɹ ɧɟɡɚɥɟɠɧɢɯ ɩɨɞɿɣ
Ⱥ ɿ ȼ ɭɦɨɜɧɚ ɣɦɨɜɿɪɧɿɫɬɶ ɩɟɪɟɬɜɨɪɸɽɬɶɫɹ ɧɚ ɡɜɢɱɚɣɧɭ:
)()|( A
P
BA
P
, ɚɛɨ
)()|( B
P
AB
P
.
Ɉɬɠɟ
ɧɟɡɚɥɟɠɧɿ ɩɨɞɿʀ (ɡɚ ɨɡɧɚɱɟɧɧɹɦ) ɧɟ ɡɦɿɧɸɸɬɶ ɣɦɨɜɿɪɧɨɫɬɿ ɩɨɹɜɢ ɿɧɲɨʀ.
Ƀɦɨɜɿɪɧɿɫɬɶ
ɞɨɛɭɬɤɭ ɡɚɥɟɠɧɢɯ ɩɨɞɿɣ Ⱥ ɿ ȼ ɜɢɡɧɚɱɚɽɬɶɫɹ ɡɚ ɮɨɪɦɭɥɨɸ:
)()|()( B
P
BA
P
ȼȺ
P
. (3.5)
ɍɦɨɜɧɚ ɣɦɨɜɿɪɧɿɫɬɶ
Ɋ(Ⱥ|ȼ) ɹɤ ɣɦɨɜɿɪɧɿɫɬɶ ɡɞɿɣɫɧɟɧɧɹ ɩɨɞɿʀ Ⱥ ɡɚ ɭɦɨɜɢ,
ɳɨ ɩɨɞɿɹ
ȼ ɜɿɞɛɭɥɚɫɹ, ɬɨɛɬɨ P(ȼ) > 0, ɜɢɡɧɚɱɚɽɬɶɫɹ ɡ (3.5):
)(
)(
)|(
BP
ȼȺP
BAP
. (3.6)
85
Ⱦɥɹ ɧɟɡɚɥɟɠɧɢɯ ɩɨɞɿɣ ɮɨɪɦɭɥɚ ɫɩɪɨɳɭɽɬɶɫɹ ɿ ɩɪɢɣɦɚɽ ɜɠɟ ɜɿɞɨɦɢɣ ɜɢɝɥɹɞ:
)()()( B
P
A
P
ȼȺ
P
.
ɉɪɢɤɥɚɞ 3.4. ȼ ɚɤɚɞɟɦɿɱɧɿɣ ɝɪɭɩɿ 15 ɯɥɨɩɰɿɜ ɿ 10 ɞɿɜɱɚɬ. əɤɚ ɣɦɨɜɿɪɧɿɫɬɶ
ɬɨɝɨ, ɳɨ ɞɜɨɽ ɧɚɜɦɚɧɧɹ ɿ ɩɿɞɪɹɞ ɜɢɛɪɚɧɢɯ ɫɬɭɞɟɧɬɿɜ ɜɢɹɜɥɹɬɶɫɹ ɞɿɜɱɚɬɚɦɢ?
Ɋɿɲɟɧɧɹ: Ɂɚɝɚɥɶɧɚ ɛɚɠɚɧɚ ɩɨɞɿɹ Ⱥ (ɜɢɛɿɪ ɧɚɜɦɚɧɧɹ ɞɜɨɯ ɫɬɭɞɟɧɬɿɜ-ɞɿɜɱɚɬ)
ɫɤɥɚɞɚɽɬɶɫɹ ɡ ɞɨɛɭɬɤɭ ɞɜɨɯ ɩɨɞɿɣ: ȼ
1
(ɜɢɩɚɞɤɨɜɢɣ ɜɢɛɿɪ ɨɞɧɿɽʀ ɞɿɜɱɢɧɢ) ɿ ȼ
2
(ɜɢɩɚɞɤɨɜɢɣ ɜɢɛɿɪ ɳɟ ɨɞɧɿɽʀ ɞɿɜɱɢɧɢ), ɬɨɛɬɨ )()(
21
BBPȺP . Ƀɦɨɜɿɪɧɿɫɬɶ
ɩɨɞɿʀ ȼ
1
– Ɋ(ȼ
1
). ɇɚɫɬɚɧɧɹ ɩɨɞɿʀ ȼ
2
ɜɿɞɛɭɜɚɽɬɶɫɹ ɩɿɫɥɹ ɩɨɞɿʀ ȼ
1
ɿ ɨɰɿɧɸɽɬɶɫɹ
ɭɦɨɜɧɨɸ ɣɦɨɜɿɪɧɿɫɬɸ
)|(
12
BBP
. Ƀɦɨɜɿɪɧɿɫɬɶ ɞɨɛɭɬɤɭ ɡɚɥɟɠɧɢɯ ɩɨɞɿɣ ȼ
1
ɿ ȼ
2
ɞɨɪɿɜɧɸɽ:
)|()()(
12121
BBPBPBBP .
Ƀɦɨɜɿɪɧɿɫɬɶ ɩɨɞɿʀ ȼ
1
ɜɢɡɧɚɱɚɽɬɶɫɹ ɹɤ ɜɿɞɧɨɲɟɧɧɹ ɤɿɥɶɤɨɫɬɿ ɞɿɜɱɚɬ (10) ɞɨ
ɡɚɝɚɥɶɧɨʀ ɤɿɥɶɤɨɫɬɿ ɫɬɭɞɟɧɬɿɜ (10+15), ɬɨɛɬɨ
4,0
1510
10
)(
1
BP
.
ɇɚɫɬɚɧɧɹ ɩɨɞɿʀ ȼ
1
ɡɦɿɧɸɽ ɭɦɨɜɢ ɞɥɹ ɨɰɿɧɤɢ ɣɦɨɜɿɪɧɨɫɬɿ ɩɨɞɿʀ ȼ
2
, ɚ ɫɚɦɟ:
ɡɦɟɧɲɭɽɬɶɫɹ ɿ ɤɿɥɶɤɿɫɬɶ ɞɿɜɱɚɬ (10–1)=9, ɿ ɡɚɝɚɥɶɧɚ ɤɿɥɶɤɿɫɬɶ ɫɬɭɞɟɧɬɿɜ
(9+15)=24. Ɍɨɞɿ ɣɦɨɜɿɪɧɿɫɬɶ ɩɨɞɿʀ ȼ
2
ɹɤ ɜɿɞɧɨɲɟɧɧɹ ɤɿɥɶɤɨɫɬɿ ɞɿɜɱɚɬ (9) ɞɨ ɡɚ-
ɝɚɥɶɧɨʀ ɤɿɥɶɤɨɫɬɿ ɫɬɭɞɟɧɬɿɜ (24) ɞɨɪɿɜɧɸɜɚɬɢɦɟ
375,0
159
9
)|(
12
BBP
.
Ɂɜɿɞɫɢ ɣɦɨɜɿɪɧɿɫɬɶ Ɋ(Ⱥ) ɩɨɞɿʀ Ⱥ ɫɤɥɚɞɚɬɢɦɟ
%1515,0375,04,0)|()()()(
12121
BBPBPBBPȺɊ
.
ȼɿɞɩɨɜɿɞɶ: ɣɦɨɜɿɪɧɿɫɬɶ ɜɢɛɨɪɭ ɧɚɜɦɚɧɧɹ ɞɜɨɯ ɫɬɭɞɟɧɬɿɜ-ɞɿɜɱɚɬ ɫɬɚɧɨɜɢɬɶ 15%.
Ɏɨɪɦɭɥɚ ɩɨɜɧɨʀ ɣɦɨɜɿɪɧɨɫɬɿ
ɉɪɢɩɭɫɬɢɦɨ, ɳɨ ɩɪɨɫɬɿɪ ɩɨɞɿɣ (ɦɧɨɠɢɧɚ ȍ) ɫɤɥɚɞɚɽɬɶɫɹ ɡ n ɩɨɩɚɪɧɨ ɧɟ-
ɫɭɦɿɫɧɢɯ ɩɨɞɿɣ – ɩɿɞɦɧɨɠɢɧ
n
HHHH ,,,,
321
!
(ɞɢɜ. ɪɢɫ. 3.4).
Ɋɢɫ. 3.4. ɉɪɨɫɬɿɪ ɩɨɞɿɣ {
n
HHH ,,,
21
!
}
ȍ
86
Ɂ ɪɢɫ. 3.4 ɜɢɞɧɨ, ɳɨ ɩɨɞɿɸ Ⱥ ɦɨɠɧɚ ɭɹɜɢɬɢ ɹɤ ɫɭɦɭ ɞɨɛɭɬɤɿɜ ɩɨɞɿʀ Ⱥ ɿ ɤɨ-
ɠɧɨʀ ɡ ɩɨɞɿɣ
n
HHH ,,,
21
! (ɫɭɦɚ ɩɟɪɟɬɢɧɿɜ ɩɿɞɦɧɨɠɢɧ Ⱥ ɡ ɩɿɞɦɧɨɠɢɧɚɦɢ ɇ
ɿ
):
Ⱥ=Ⱥ·ɇ
1
+Ⱥ·ɇ
2
+...+Ⱥ·ɇ
n
=
¦
n
ɿ
ɿ
ɇȺ
1
.
Ƀɦɨɜɿɪɧɿɫɬɶ ɩɨɞɿʀ Ⱥ (ɡɚ ɬɟɨɪɟɦɨɸ ɞɨɞɚɜɚɧɧɹ ɣɦɨɜɿɪɧɨɫɬɟɣ) ɜɢɡɧɚɱɢɬɶɫɹ
)()(
1
¦
n
ɿ
ɿ
ɇȺPAP .
Ƀɦɨɜɿɪɧɿɫɬɶ ɩɨɞɿʀ Ⱥ (ɡɚ ɬɟɨɪɟɦɨɸ ɦɧɨɠɟɧɧɹ ɣɦɨɜɿɪɧɨɫɬɟɣ) ɜɢɡɧɚɱɢɬɶɫɹ:
)()|()(
1
i
n
i
i
HPHAPAP
¦
. (3.7)
ȼɢɪɚɡ (3.7) ɽ
ɮɨɪɦɭɥɨɸ ɩɨɜɧɨʀ ɣɦɨɜɿɪɧɨɫɬɿ Ɋ(Ⱥ) ɩɨɞɿʀ Ⱥ, ɹɤɳɨ ɩɨɞɿɹ Ⱥ
ɡɚɥɟɠɢɬɶ ɜɿɞ ɫɢɫɬɟɦɢ ɩɨɞɿɣ. ɉɨɞɿʀ
n
HHH ,,,
21
!
ɩɪɢɣɧɹɬɨ ɧɚɡɢɜɚɬɢ ɝɿɩɨɬɟɡɚ-
ɦɢ, ɡɚ ɹɤɢɦɢ ɦɨɠɟ ɜɿɞɛɭɬɢɫɹ ɩɨɞɿɹ Ⱥ.
ɉɪɢɤɥɚɞ 3.5. ɍ ɬɪɶɨɯ ɝɪɭɩɚɯ ɫɬɭɞɟɧɬɿɜ (ɱɢɫɟɥɶɧɿɫɬɸ 28, 20 ɿ 25 ɨɫɿɛ) ɜɿɞ-
ɦɿɧɧɢɤɢ ɫɬɚɧɨɜɥɹɬɶ 7, 2 ɿ 5 ɫɬɭɞɟɧɬɿɜ ɜɿɞɩɨɜɿɞɧɨ. əɤɚ ɣɦɨɜɿɪɧɿɫɬɶ ɬɨɝɨ, ɳɨ
ɧɚɜɦɚɧɧɹ ɜɢɛɪɚɧɢɣ ɫɬɭɞɟɧɬ ɽ ɜɿɞɦɿɧɧɢɤɨɦ?
Ɋɿɲɟɧɧɹ: ɉɪɨɫɬɿɪ ɩɨɞɿɣ (ɦɧɨɠɢɧɚ ȍ) ɫɤɥɚɞɚɽɬɶɫɹ ɡ 3-ɯ ɩɨɩɚɪɧɨ ɧɟɫɭɦɿɫ-
ɧɢɯ ɩɨɞɿɣ – ɩɿɞɦɧɨɠɢɧ
321
,, HHH (ɞɢɜ. ɪɢɫ. 3.5). ɉɨɞɿɹ Ⱥ – ɰɟ ɪɟɡɭɥɶɬɚɬ ɜɢ-
ɩɪɨɛɭɜɚɧɧɹ – ɜɢɛɨɪɭ ɧɚɜɦɚɧɧɹ ɫɬɭɞɟɧɬɚ-ɜɿɞɦɿɧɧɢɤɚ ɡ ɣɦɨɜɿɪɧɿɫɬɸ Ɋ(Ⱥ).
Ɋɢɫ. 3.5. ɉɪɨɫɬɿɪ ɩɨɞɿɣ {
321
,, HHH
}
ȍ
Ƀɦɨɜɿɪɧɿɫɬɶ Ɋ(Ⱥ) ɪɨɡɪɚɯɨɜɭɽɬɶɫɹ ɡɚ ɮɨɪɦɭɥɨɸ ɩɨɜɧɨʀ ɣɦɨɜɿɪɧɨɫɬɿ (n=3):
)()|()()|()()|()()|()(
332211
3
1
ɇPɇAPɇPɇAPɇPɇAPɇPɇAPAP
i
i
i
¦
.
Ƀɦɨɜɿɪɧɿɫɬɶ ɝɿɩɨɬɟɡɢ ɇ
1
(ɩɨɞɿʀ ɩɪɨ ɬɟ, ɳɨ ɜɿɞɦɿɧɧɢɤ ɜɢɛɪɚɧɢɣ ɡ 1-ʀ ɝɪɭɩɢ ):
38,0
73
28
252028
28
)(
1
|
ɇP .
87
Ƀɦɨɜɿɪɧɿɫɬɶ ɝɿɩɨɬɟɡɢ ɇ
2
(ɩɨɞɿʀ ɩɪɨ ɬɟ, ɳɨ ɜɿɞɦɿɧɧɢɤ ɜɢɛɪɚɧɢɣ ɡ 2-ʀ ɝɪɭɩɢ):
27,0
73
20
252028
20
)(
2
|
ɇP .
Ƀɦɨɜɿɪɧɿɫɬɶ ɝɿɩɨɬɟɡɢ ɇ
3
(ɩɨɞɿʀ ɩɪɨ ɬɟ, ɳɨ ɜɿɞɦɿɧɧɢɤ ɜɢɛɪɚɧɢɣ ɡ 3-ʀɝɪɭɩɢ):
34,0
73
25
252028
25
)(
3
|
ɇP .
ɍɦɨɜɧɿ ɣɦɨɜɿɪɧɨɫɬɿ ɩɨɞɿʀ Ⱥ ɞɥɹ ɤɨɠɧɨʀ ɝɿɩɨɬɟɡɢ ɪɨɡɪɚɯɨɜɭɸɬɶɫɹ ɹɤ:
25,0
28
7
)|(
1
ɇAP
;
10,0
20
2
)|(
2
ɇAP
;
20,0
25
5
)|(
3
ɇAP
.
Ɂɚ ɮɨɪɦɭɥɨɸ ɩɨɜɧɨʀ ɣɦɨɜɿɪɧɨɫɬɿ Ɋ(Ⱥ) ɪɨɡɪɚɯɨɜɭɽɬɶɫɹ ɹɤ:
)()|()()|()()|()(
332211
ɇPɇAPɇPɇAPɇPɇAPAP
ɚɛɨ
19,0068,0027,0095,034,020,027,010,038,025,0)( ||
A
P
.
ȼɿɞɩɨɜɿɞɶ: ɣɦɨɜɿɪɧɿɫɬɶ Ɋ(Ⱥ) ɩɨɞɿʀ Ⱥ ɩɪɨ ɬɟ, ɳɨ ɧɚɜɦɚɧɧɹ ɜɢɛɪɚɧɢɣ ɡ ɬɪɶɨɯ
ɝɪɭɩ ɫɬɭɞɟɧɬ ɽ ɜɿɞɦɿɧɧɢɤɨɦ, ɫɤɥɚɞɚɽ ɩɪɢɛɥɢɡɧɨ 0,19 ɚɛɨ 19%.
ɉɪɢɤɥɚɞ 3.6. ȼ ɚɤɚɞɟɦɿɱɧɿɣ ɝɪɭɩɿ ɡ 10 ɫɬɭɞɟɧɬɿɜ, ɳɨ ɩɪɢɣɲɥɢ ɧɚ ɟɤɡɚɦɟɧ,
3 ɫɬɭɞɟɧɬɢ ɩɿɞɝɨɬɨɜɥɟɧɿ ɜɿɞɦɿɧɧɨ, 4 – ɞɨɛɪɟ, 2 – ɡɚɞɨɜɿɥɶɧɨ ɿ 1 – ɩɨɝɚɧɨ. ȼɿɞ-
ɦɿɧɧɨ ɩɿɞɝɨɬɨɜɥɟɧɢɣ ɫɬɭɞɟɧɬ ɦɨɠɟ ɜɿɞɩɨɜɿɫɬɢ ɧɚ ɜɫɿ 20 ɩɢɬɚɧɶ, ɞɨɛɪɟ ɩɿɞɝɨ-
ɬɨɜɥɟɧɢɣ – ɧɚ 16, ɡɚɞɨɜɿɥɶɧɨ – ɧɚ 10 ɩɢɬɚɧɶ, ɩɨɝɚɧɨ – ɧɚ 5 ɩɢɬɚɧɶ. əɤɚ ɣɦɨɜɿ-
ɪɧɿɫɬɶ ɬɨɝɨ, ɳɨ ɧɚɜɦɚɧɧɹ ɜɢɤɥɢɤɚɧɢɣ ɡ ɰɿɽʀ ɝɪɭɩɢ ɫɬɭɞɟɧɬ ɜɿɞɩɨɜɿɫɬɶ ɧɚ ɞɜɚ
ɡɚɞɚɧɢɯ ɩɢɬɚɧɧɹ?
Ɋɿɲɟɧɧɹ: ɉɪɨɫɬɿɪ ɩɨɞɿɣ (ɦɧɨɠɢɧɚ ȍ) ɫɤɥɚɞɚɽɬɶɫɹ ɡ 4-ɯ ɧɟɫɭɦɿɫɧɢɯ ɩɨɞɿɣ –
ɩɿɞɦɧɨɠɢɧ
4321
,,, HHHH
(ɞɢɜ ɪɢɫ. 3.6).
Ɋɢɫ. 3.6. ɉɪɨɫɬɿɪ ɩɨɞɿɣ {
4321
,,, HHHH
}
ȍ
ɉɨɞɿɹ Ⱥ – ɰɟ ɪɟɡɭɥɶɬɚɬ ɜɢɩɪɨɛɭɜɚɧɧɹ – ɜɢɛɨɪɭ ɧɚɜɦɚɧɧɹ ɫɬɭɞɟɧɬɚ, ɹɤɢɣ
ɭɫɩɿɲɧɨ ɜɿɞɩɨɜɿɫɬɶ ɧɚ ɞɜɚ ɡɚɞɚɧɢɯ ɩɢɬɚɧɧɹ. Ɋ(Ⱥ) – ɣɦɨɜɿɪɧɿɫɬɶ ɰɿɽʀ ɩɨɞɿʀ ɪɨɡ-
88
ɪɚɯɨɜɭɽɬɶɫɹ ɡɚ ɮɨɪɦɭɥɨɸ ɩɨɜɧɨʀ ɣɦɨɜɿɪɧɨɫɬɿ (n=4):
)()|()(
4
1
i
i
i
ɇPɇAPAP
¦
.
ȼɢɫɭɜɚɽɦɨ ɱɨɬɢɪɢ ɝɿɩɨɬɟɡɢ ɳɨɞɨ ɣɦɨɜɿɪɧɨɫɬɿ ɩɨɹɜɢ (ɭ ɪɟɡɭɥɶɬɚɬɿ ɜɢɤɥɢɤɭ
ɧɚɜɦɚɧɧɹ) ɬɨɝɨ ɱɢ ɿɧɲɨɝɨ ɫɬɭɞɟɧɬɚ ɡ ɩɟɜɧɨɸ ɩɿɞɝɨɬɨɜɤɨɸ:
– ɝɿɩɨɬɟɡɚ ɇ
1
: ɰɟ ɛɭɞɟ ɫɬɭɞɟɧɬ, ɳɨ ɩɿɞɝɨɬɨɜɥɟɧɢɣ ɜɿɞɦɿɧɧɨ, ɣɦɨɜɿɪɧɿɫɬɶ
ɣɨɝɨ ɩɨɹɜɢ Ɋ(ɇ
1
) = 3/10 = 0,3;
– ɝɿɩɨɬɟɡɚ ɇ
2
: ɰɟ ɛɭɞɟ ɫɬɭɞɟɧɬ, ɳɨ ɩɿɞɝɨɬɨɜɥɟɧɢɣ ɞɨɛɪɟ, ɣɦɨɜɿɪɧɿɫɬɶ ɣɨ-
ɝɨ ɩɨɹɜɢ Ɋ(ɇ
2
) = 4/10 =0,4;
– ɝɿɩɨɬɟɡɚ ɇ
3
: ɰɟ ɛɭɞɟ ɫɬɭɞɟɧɬ, ɳɨ ɩɿɞɝɨɬɨɜɥɟɧɢɣ ɡɚɞɨɜɿɥɶɧɨ, ɣɦɨɜɿɪɧɿɫɬɶ
ɣɨɝɨ ɩɨɹɜɢ Ɋ(ɇ
3
) = 2/10 = 0,2;
– ɝɿɩɨɬɟɡɚ ɇ
4
: ɰɟ ɛɭɞɟ ɫɬɭɞɟɧɬ, ɳɨ ɩɿɞɝɨɬɨɜɥɟɧɢɣ ɩɨɝɚɧɨ, ɣɦɨɜɿɪɧɿɫɬɶ
ɣɨɝɨ ɩɨɹɜɢ Ɋ(ɇ
4
) = 1/10 = 0,1.
ɍɦɨɜɧɿ ɣɦɨɜɿɪɧɨɫɬɿ ɜɢɤɨɧɚɧɧɹ ɞɜɨɯ ɡɚɜɞɚɧɶ ɫɬɭɞɟɧɬɿɜ ɡ ɩɟɜɧɨɸ ɩɿɞɝɨɬɨ-
ɜɤɨɸ ɪɨɡɪɚɯɨɜɭɸɬɶɫɹ ɹɤ ɣɦɨɜɿɪɧɨɫɬɿ ɞɨɛɭɬɤɭ ɞɜɨɯ ɡɚɥɟɠɧɢɯ ɩɨɞɿɣ (ɩɨɞɿɣ
ɭɫɩɿɲɧɨɝɨ ɜɢɤɨɧɚɧɧɹ ɞɜɨɯ ɡɚɜɞɚɧɶ). Ɂɝɿɞɧɨ ɡ ɬɟɨɪɟɦɨɸ ɦɧɨɠɟɧɧɹ:
– ɭɦɨɜɧɚ ɣɦɨɜɿɪɧɿɫɬɶ ɜɢɤɨɧɚɧɧɹ ɞɜɨɯ ɡɚɜɞɚɧɶ ɫɬɭɞɟɧɬɨɦ, ɳɨ ɩɿɞɝɨɬɨɜɥɟ-
ɧɢɣ ɜɿɞɦɿɧɧɨ, ɞɨɪɿɜɧɸɜɚɬɢɦɟ Ɋ(Ⱥ|ɇ
1
) = (20/20)ǜ(19/19) = 1;
– ɭɦɨɜɧɚ ɣɦɨɜɿɪɧɿɫɬɶ ɜɢɤɨɧɚɧɧɹ ɞɜɨɯ ɡɚɜɞɚɧɶ ɫɬɭɞɟɧɬɨɦ, ɳɨ ɩɿɞɝɨɬɨɜɥɟ-
ɧɢɣ ɞɨɛɪɟ, ɞɨɪɿɜɧɸɜɚɬɢɦɟ Ɋ(Ⱥ|ɇ
2
) = (16/20)ǜ(15/19) § 0,63;
– ɭɦɨɜɧɚ ɣɦɨɜɿɪɧɿɫɬɶ ɜɢɤɨɧɚɧɧɹ ɞɜɨɯ ɡɚɜɞɚɧɶ ɫɬɭɞɟɧɬɨɦ, ɳɨ ɩɿɞɝɨɬɨɜɥɟ-
ɧɢɣ ɡɚɞɨɜɿɥɶɧɨ, ɞɨɪɿɜɧɸɜɚɬɢɦɟ Ɋ(Ⱥ|ɇ
3
) = (10/20)ǜ(9/19) § 0,24;
– ɭɦɨɜɧɚ ɣɦɨɜɿɪɧɿɫɬɶ ɜɢɤɨɧɚɧɧɹ ɞɜɨɯ ɡɚɜɞɚɧɶ ɫɬɭɞɟɧɬɨɦ, ɳɨ ɩɿɞɝɨɬɨɜɥɟ-
ɧɢɣ ɩɨɝɚɧɨ, ɞɨɪɿɜɧɸɜɚɬɢɦɟ Ɋ(Ⱥ|ɇ
4
) = (5/20)ǜ(4/19) § 0,053.
Ɂɚ ɮɨɪɦɭɥɨɸ ɩɨɜɧɨʀ ɣɦɨɜɿɪɧɨɫɬɿ Ɋ(Ⱥ) ɪɨɡɪɚɯɨɜɭɽɬɶɫɹ ɹɤ:
)()|()()|()()|()()|()(
44332211
ɇPɇAPɇPɇAPɇPɇAPɇPɇAPAP
ɚɛɨ
Ɋ(Ⱥ) = 1·0,3+0,63·0,4+0,24·0,2+0,053·0,1 § 0,605 § 60,5%.
ȼɿɞɩɨɜɿɞɶ: ɣɦɨɜɿɪɧɿɫɬɶ Ɋ(Ⱥ) ɩɨɞɿʀ Ⱥ ɩɪɨ ɬɟ, ɳɨ ɜɢɤɥɢɤɚɧɢɣ ɧɚɜɦɚɧɧɹ ɫɬɭ-
ɞɟɧɬ ɜɿɞɩɨɜɿɫɬɶ ɧɚ ɞɜɚ ɡɚɞɚɧɢɯ ɩɢɬɚɧɧɹ, ɫɤɥɚɞɚɽ ɩɪɢɛɥɢɡɧɨ 0,605 ɚɛɨ 60,5%.
89
Ɏɨɪɦɭɥɚ Ȼɚɣɽɫɚ
Ɏɨɪɦɭɥɚ ɩɨɜɧɨʀ ɣɦɨɜɿɪɧɨɫɬɿ ɞɚɽ ɦɨɠɥɢɜɿɫɬɶ ɪɨɡɪɚɯɭɜɚɬɢ ɣɦɨɜɿɪɧɿɫɬɶ
Ɋ(Ⱥ) ɩɨɞɿʀ Ⱥ, ɹɤɳɨ ɜɨɧɚ ɡɚɥɟɠɢɬɶ ɜɿɞ ɫɢɫɬɟɦɢ ɩɨɞɿɣ-ɝɿɩɨɬɟɡ
n
HHH ,,,
21
!
, ɡɚ
ɭɦɨɜɧɢɦɢ ɣɦɨɜɿɪɧɨɫɬɹɦɢ ɹɤɢɯ
)|(
1
ɇAP
,
)|(
2
ɇAP ,
!
, )|(
n
ɇAP ɦɨɠɟ ɜɿɞɛɭ-
ɬɢɫɹ ɰɹ ɩɨɞɿɹ Ⱥ. ɉɪɨɬɟ ɜɚɠɥɢɜɢɦ ɡɚɜɞɚɧɧɹɦ ɦɚɬɟɦɚɬɢɤɢ ɽ ɪɨɡɪɚɯɭɧɤɢ ɭɦɨɜ-
ɧɨʀ ɣɦɨɜɿɪɧɨɫɬɿ
)|( AɇP
ɿ
ɝɿɩɨɬɟɡɢ ɇ
ɿ
, ɹɤɳɨ ɜɿɞɨɦɨ, ɳɨ ɭ ɜɢɩɪɨɛɭɜɚɧɧɿ ɩɨɞɿɹ
Ⱥ ɜɠɟ ɜɿɞɛɭɥɚɫɹ. Ɂɝɿɞɧɨ ɡ ɬɟɨɪɟɦɨɸ ɦɧɨɠɟɧɧɹ ɣɦɨɜɿɪɧɨɫɬɟɣ ɦɨɠɧɚ ɡɚɩɢɫɚɬɢ
Ɋ(ɇ
ɿ
· Ⱥ) = Ɋ(ɇ
ɿ
) · Ɋ(Ⱥ | ɇ
ɿ
) = Ɋ(Ⱥ) · Ɋ(ɇ
ɿ
| Ⱥ).
Ɂɜɿɞɫɢ
)(
)|()(
)|(
AP
ɇAPɇP
AɇP
ii
i
. (3.8)
əɤɳɨ ɡɧɚɦɟɧɧɢɤ Ɋ(Ⱥ) ɡɚɦɿɧɢɬɢ ɮɨɪɦɭɥɨɸ ɩɨɜɧɨʀ ɣɦɨɜɿɪɧɨɫɬɿ (3.7), ɨɬɪɢ-
ɦɚɽɦɨ
ɮɨɪɦɭɥɭ Ȼɚɣɽɫɚ:
¦
n
i
ii
ii
i
HAPHP
ɇAPɇP
AɇP
1
)|()(
)|()(
)|(
, (3.9)
ɞɟ
n
ɇɇɇ ,,,
21
! – ɩɨɩɚɪɧɨ ɧɟɫɭɦɿɫɧɿ ɩɨɞɿʀ, ɳɨ ɭɬɜɨɪɸɸɬɶ ɩɨɜɧɭ ɝɪɭɩɭ.
Ɏɨɪɦɭɥɚ Ȼɚɣɽɫɚ ɞɚɽ ɦɨɠɥɢɜɿɫɬɶ ɩɿɞɪɚɯɭɜɚɬɢ «ɚɩɨɫɬɟɪɿɨɪɧɿ»
10
ɣɦɨɜɿɪɧɨɫɬɿ
Ɋ(ɇ
ɿ
| Ⱥ) ɡɚ ɞɨɩɨɦɨɝɨɸ «ɚɩɪɿɨɪɧɢɯ»
11
ɣɦɨɜɿɪɧɨɫɬɟɣ Ɋ(ɇ
ɿ
) «ɝɿɩɨɬɟɡ» ɇ
ɿ
.
ɉɪɢɤɥɚɞ 3.7. Ɂɚ ɭɦɨɜɚɦɢ ɩɪɢɤɥɚɞɭ 3.6 ɜɢɤɥɢɤɚɧɢɣ ɧɚɜɦɚɧɧɹ ɫɬɭɞɟɧɬ ɜɿɞ-
ɩɨɜɿɜ ɧɚ ɬɪɢ ɡɚɞɚɧɢɯ ɩɢɬɚɧɧɹ. əɤɚ ɣɦɨɜɿɪɧɿɫɬɶ ɬɨɝɨ, ɳɨ ɰɟɣ ɫɬɭɞɟɧɬ ɽ: ɚ) ɜɿɞ-
ɦɿɧɧɨ ɩɿɞɝɨɬɨɜɥɟɧɢɣ; ɛ) ɩɿɞɝɨɬɨɜɥɟɧɢɣ ɩɨɝɚɧɨ?
Ɋɿɲɟɧɧɹ: ȼɢɫɭɜɚɽɦɨ ɱɨɬɢɪɢ ɝɿɩɨɬɟɡɢ ɳɨɞɨ ɣɦɨɜɿɪɧɨɫɬɿ ɩɨɹɜɢ (ɭ ɪɟɡɭɥɶɬɚ-
ɬɿ ɜɢɤɥɢɤɭ ɧɚɜɦɚɧɧɹ) ɬɨɝɨ ɱɢ ɿɧɲɨɝɨ ɫɬɭɞɟɧɬɚ ɡ ɩɟɜɧɨɸ ɩɿɞɝɨɬɨɜɤɨɸ:
– ɝɿɩɨɬɟɡɚ ɇ
1
: ɰɟ ɛɭɜ ɫɬɭɞɟɧɬ, ɳɨ ɩɿɞɝɨɬɨɜɥɟɧɢɣ ɜɿɞɦɿɧɧɨ, ɣɦɨɜɿɪɧɿɫɬɶ
ɣɨɝɨ ɩɨɹɜɢ Ɋ(ɇ
1
) = 3/10 = 0,3;
– ɝɿɩɨɬɟɡɚ ɇ
2
: ɰɟ ɛɭɜ ɫɬɭɞɟɧɬ, ɳɨ ɩɿɞɝɨɬɨɜɥɟɧɢɣ ɞɨɛɪɟ, ɣɦɨɜɿɪɧɿɫɬɶ ɣɨɝɨ
ɩɨɹɜɢ Ɋ(ɇ
2
) = 4/10 = 0,4;
10
a posteriori (ÏÆÖ.) – ÑÆ ÒÕÑÒÈ ÉÒÕÏɾ.
11
a priori (ÏÆÖ.) – ÉÒ ÉÒÕÏɾ.
90
– ɝɿɩɨɬɟɡɚ ɇ
3
: ɰɟ ɛɭɜ ɫɬɭɞɟɧɬ, ɳɨ ɩɿɞɝɨɬɨɜɥɟɧɢɣ ɡɚɞɨɜɿɥɶɧɨ, ɣɦɨɜɿɪɧɿɫɬɶ
ɣɨɝɨ ɩɨɹɜɢ Ɋ(ɇ
3
) = 2/10 = 0,2;
– ɝɿɩɨɬɟɡɚ ɇ
4
: ɰɟ ɛɭɜ ɫɬɭɞɟɧɬ, ɳɨ ɩɿɞɝɨɬɨɜɥɟɧɢɣ ɩɨɝɚɧɨ, ɣɦɨɜɿɪɧɿɫɬɶ ɣɨ-
ɝɨ ɩɨɹɜɢ Ɋ(ɇ
4
) = 1/10 = 0,1.
ɍɦɨɜɧɿ ɣɦɨɜɿɪɧɨɫɬɿ ɜɢɤɨɧɚɧɧɹ ɬɪɶɨɯ ɡɚɜɞɚɧɶ ɬɨɝɨ ɱɢ ɿɧɲɨɝɨ ɫɬɭɞɟɧɬɚ ɡ
ɩɟɜɧɨɸ ɩɿɞɝɨɬɨɜɤɨɸ ɪɨɡɪɚɯɨɜɭɸɬɶɫɹ ɹɤ ɣɦɨɜɿɪɧɨɫɬɿ ɞɨɛɭɬɤɭ ɬɪɶɨɯ ɡɚɥɟɠɧɢɯ
ɩɨɞɿɣ (ɭɫɩɿɲɧɨɝɨ ɜɢɤɨɧɚɧɧɹ ɬɪɶɨɯ ɡɚɜɞɚɧɶ). Ɂɝɿɞɧɨ ɡ ɬɟɨɪɟɦɨɸ ɦɧɨɠɟɧɧɹ:
– ɭɦɨɜɧɚ ɣɦɨɜɿɪɧɿɫɬɶ ɜɢɤɨɧɚɧɧɹ ɬɪɶɨɯ ɡɚɜɞɚɧɶ ɫɬɭɞɟɧɬɨɦ, ɳɨ ɩɿɞɝɨɬɨɜɥɟ-
ɧɢɣ ɜɿɞɦɿɧɧɨ, ɞɨɪɿɜɧɸɜɚɬɢɦɟ Ɋ(Ⱥ|ɇ
1
) = (20/20)ǜ(19/19)ǜ(18/18) = 1;
– ɭɦɨɜɧɚ ɣɦɨɜɿɪɧɿɫɬɶ ɜɢɤɨɧɚɧɧɹ ɬɪɶɨɯ ɡɚɜɞɚɧɶ ɫɬɭɞɟɧɬɨɦ, ɳɨ ɩɿɞɝɨɬɨɜɥɟ-
ɧɢɣ ɞɨɛɪɟ, ɞɨɪɿɜɧɸɜɚɬɢɦɟ Ɋ(Ⱥ|ɇ
2
) = (16/20)ǜ(15/19)ǜ(14/18) § 0,491;
– ɭɦɨɜɧɚ ɣɦɨɜɿɪɧɿɫɬɶ ɜɢɤɨɧɚɧɧɹ ɬɪɶɨɯ ɡɚɜɞɚɧɶ ɫɬɭɞɟɧɬɨɦ, ɳɨ ɩɿɞɝɨɬɨɜɥɟ-
ɧɢɣ ɡɚɞɨɜɿɥɶɧɨ, ɞɨɪɿɜɧɸɜɚɬɢɦɟ Ɋ(Ⱥ|ɇ
3
) = (10/20)ǜ(9/19)ǜ(8/18) § 0,105;
– ɭɦɨɜɧɚ ɣɦɨɜɿɪɧɿɫɬɶ ɜɢɤɨɧɚɧɧɹ ɬɪɶɨɯ ɡɚɜɞɚɧɶ ɫɬɭɞɟɧɬɨɦ, ɳɨ ɩɿɞɝɨɬɨɜɥɟ-
ɧɢɣ ɩɨɝɚɧɨ, ɞɨɪɿɜɧɸɜɚɬɢɦɟ Ɋ(Ⱥ|ɇ
4
) = (5/20)ǜ(4/19)ǜ(3/18) § 0,009.
Ɂɚ ɮɨɪɦɭɥɨɸ Ȼɚɣɽɫɚ:
ɚ) ɣɦɨɜɿɪɧɿɫɬɶ ɬɨɝɨ, ɳɨ ɰɟ ɛɭɜ ɫɬɭɞɟɧɬ, ɩɿɞɝɨɬɨɜɥɟɧɢɣ ɜɿɞɦɿɧɧɨ, ɫɤɥɚɞɚɽ
¦
4
1
11
1
)|()(
)|()(
)|(
i
ii
HAPHP
ɇAPɇP
AɇP
,
ɚɛɨ
%5858,0
009,01,0105,02,0491,04,013,0
13,0
)|(
1
||
AɇP
;
ɛ) ɣɦɨɜɿɪɧɿɫɬɶ ɬɨɝɨ, ɳɨ ɰɟ ɛɭɜ ɫɬɭɞɟɧɬ, ɩɿɞɝɨɬɨɜɥɟɧɢɣ ɩɨɝɚɧɨ, ɫɤɥɚɞɚɽ
%2,0002,0
009,01,0105,02,0491,04,013,0
009,01,0
)|(
2
||
AɇP
.
ȼɿɞɩɨɜɿɞɶ: ɣɦɨɜɿɪɧɿɫɬɶ ɬɨɝɨ, ɳɨ ɧɚ ɜɫɿ ɬɪɢ ɩɢɬɚɧɧɹ ɞɚɜ ɜɿɞɩɨɜɿɞɶ ɜɿɞɦɿɧɧɨ
ɩɿɞɝɨɬɨɜɥɟɧɢɣ ɫɬɭɞɟɧɬ, ɞɨɪɿɜɧɸɽ 58%, ɭ ɬɨɣ ɱɚɫ ɹɤ ɣɦɨɜɿɪɧɿɫɬɶ ɞɥɹ ɩɨɝɚɧɨ
ɩɿɞɝɨɬɨɜɥɟɧɨɝɨ ɫɤɥɚɞɚɽ ɥɢɲɟ 0,2%. Ɉɬɪɢɦɚɧɢɣ ɪɟɡɭɥɶɬɚɬ ɬɚɤɨɠ ɦɨɠɟ ɨɡɧɚ-
ɱɚɬɢ, ɳɨ ɩɪɨɰɟɞɭɪɚ ɿɫɩɢɬɭ ɡɚ ɞɚɧɢɦɢ ɤɪɢɬɟɪɿɹɦɢ ɦɚɽ ɞɨɜɨɥɿ ɜɢɫɨɤɢɣ ɪɿɜɟɧɶ
ɞɿɚɝɧɨɫɬɢɱɧɢɯ ɜɥɚɫɬɢɜɨɫɬɟɣ – ɩɨɪɿɜɧɹɣɬɟ 58% ɞɥɹ ɜɿɞɦɿɧɧɢɤɚ ɿ 0,2% ɞɥɹ ɩɨ-
ɝɚɧɨ ɩɿɞɝɨɬɨɜɥɟɧɨɝɨ ɫɬɭɞɟɧɬɚ.