Математика и информатика. Решение логико-познавательных задач. Учебное пособие для студентов вузов
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D
DȺ
ǸȐș. 4.6
P(A) |
Sɩɪɹɦɨɭɝ |
3 4 |
| 0,055 . |
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Sɷɥɥɢɩɫɚ |
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Σ 7 10 |
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ɋɬɚɬɢɫɬɢɱɟɫɤɨɟ ɨɩɪɟɞɟɥɟɧɢɟ ɜɟɪɨɹɬɧɨɫɬɢ. ɉɭɫɬɶ ɢɫɩɵɬɚɧɢɟ, ɫ
ɤɨɬɨɪɵɦ ɫɜɹɡɚɧɨ ɧɟɤɨɬɨɪɨɟ ɫɨɛɵɬɢɟ Ⱥ, ɩɨɜɬɨɪɹɟɬɫɹ n ɪɚɡ, ɚ ɫɚɦɨ ɫɨ-
ɛɵɬɢɟ Ⱥ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɪɢ ɷɬɨɦ k ɪɚɡ. Ɉɬɧɨɲɟɧɢɟ kn ɩɪɢɧɹɬɨ ɧɚɡɵ-
ɜɚɬɶ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɱɚɫɬɨɬɨɣ ɫɨɛɵɬɢɹ Ⱥ.
Ⱦɨɤɚɡɚɧɨ, ɱɬɨ ɩɪɢ ɧɟɨɝɪɚɧɢɱɟɧɧɨɦ ɭɜɟɥɢɱɟɧɢɢ ɱɢɫɥɚ ɢɫɩɵɬɚɧɢɣ ( n ο φ ) ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɱɚɫɬɨɬɚ ɫɨɛɵɬɢɹ Ⱥ ɛɭɞɟɬ ɦɚɥɨ ɨɬɥɢɱɚɬɶɫɹ ɨɬ ɜɟɪɨɹɬɧɨɫɬɢ ɩɨɹɜɥɟɧɢɹ ɷɬɨɝɨ ɫɨɛɵɬɢɹ ɜ ɨɬɞɟɥɶɧɨɦ ɢɫɩɵɬɚɧɢɢ.
P(A) lim k
nοφ n
ȼɜɟɞɟɧɧɚɹ ɬɚɤɢɦ ɨɛɪɚɡɨɦ ɜɟɪɨɹɬɧɨɫɬɶ ɫɨɛɵɬɢɹ ɧɚɡɵɜɚɟɬɫɹ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ. ȼɟɪɨɹɬɧɨɫɬɶ (ɜ ɫɬɚɬɢɫɬɢɱɟɫɤɨɦ ɫɦɵɫɥɟ) ɟɫɬɶ ɱɢɫɥɨ, ɨɤɨɥɨ ɤɨɬɨɪɨɝɨ ɤɨɥɟɛɥɟɬɫɹ ɧɚɛɥɸɞɚɟɦɚɹ ɱɚɫɬɨɬɚ ɫɨɛɵɬɢɹ. ɉɪɢ ɞɨɫɬɚ-
ɬɨɱɧɨ ɛɨɥɶɲɨɦ n ɦɨɠɧɨ ɩɪɢɧɹɬɶ, ɱɬɨ P(A) | kn .
ɋɬɚɬɢɫɬɢɱɟɫɤɚɹ ɜɟɪɨɹɬɧɨɫɬɶ, ɫɨɝɥɚɫɧɨ ɬɟɨɪɟɦɟ ə.Ȼɟɪɧɭɥɥɢ, ɮɢɤɫɢɪɭɟɬ ɫɜɹɡɶ ɦɟɠɞɭ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɱɚɫɬɨɬɨɣ ɫɨɛɵɬɢɹ ɤɚɤ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɢɡɦɟɪɹɟɦɨɣ ɜɟɥɢɱɢɧɨɣ ɢ ɜɟɪɨɹɬɧɨɫɬɶɸ ɤɚɤ ɮɨɪɦɚɥɶɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ ɫɥɭɱɚɣɧɨɝɨ ɫɨɛɵɬɢɹ.
ɉɪɢɦɟɪ 42. ɋɬɪɟɥɨɤ ɫɞɟɥɚɥ 40 ɜɵɫɬɪɟɥɨɜ ɩɨ ɦɢɲɟɧɢ, ɩɪɢ ɷɬɨɦ ɛɵɥɨ ɡɚɮɢɤɫɢɪɨɜɚɧɨ 26 ɩɨɩɚɞɚɧɢɣ. Ɉɰɟɧɢɦ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɜ ɦɢɲɟɧɶ (ɫɨɛɵɬɢɟ Ⱥ) ɷɬɢɦ ɫɬɪɟɥɤɨɦ.
91
ɋɨɛɵɬɢɟ Ⱥ — ɩɨɩɚɞɚɧɢɟ ɜ ɦɢɲɟɧɶ. Ɉɛɳɟɟ ɱɢɫɥɨ ɢɫɩɵɬɚɧɢɣ n 40 . ɑɢɫɥɨ ɢɫɩɵɬɚɧɢɣ k 26 , ɜ ɤɨɬɨɪɵɯ ɧɚɫɬɭɩɢɥɨ ɫɨɛɵɬɢɟ Ⱥ. ɋɥɟɞɨɜɚɬɟɥɶ-
26
ɧɨ, P(A) 0,65 . 40
Ɋɚɫɫɦɨɬɪɟɧɧɵɟ ɜɵɲɟ ɩɪɹɦɵɟ ɦɟɬɨɞɵ ɜɵɱɢɫɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ ɧɟ ɜɫɟɝɞɚ ɷɮɮɟɤɬɢɜɧɵ. ɇɚ ɩɪɚɤɬɢɤɟ ɱɚɫɬɨ ɩɪɢɦɟɧɹɸɬ ɤɨɫɜɟɧɧɵɟ ɦɟɬɨɞɵ ɜɵɱɢɫɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ, ɩɨɡɜɨɥɹɸɳɢɟ ɧɚɯɨɞɢɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɨɞɧɢɯ ɫɨɛɵɬɢɣ ɩɨ ɢɡɜɟɫɬɧɵɦ ɜɟɪɨɹɬɧɨɫɬɹɦ ɞɪɭɝɢɯ ɫɨɛɵɬɢɣ. ȼ ɨɫɧɨɜɟ ɷɬɢɯ ɤɨɫɜɟɧɧɵɯ ɦɟɬɨɞɨɜ ɥɟɠɚɬ ɨɫɧɨɜɧɵɟ ɬɟɨɪɟɦɵ ɢ ɮɨɪɦɭɥɵ ɬɟɨɪɢɢ ɜɟɪɨɹɬɧɨɫɬɟɣ.
4.4. ǺȍȖȘȍȔȣ țȔȕȖȎȍȕȐȧ Ȑ șȓȖȎȍȕȐȧ ȊȍȘȖȧȚȕȖșȚȍȑ
ɋɨɛɵɬɢɟ ȼ ɧɚɡɵɜɚɟɬɫɹ ɡɚɜɢɫɢɦɵɦ ɨɬ ɫɨɛɵɬɢɹ Ⱥ, ɟɫɥɢ ɜɟɪɨɹɬɧɨɫɬɶ ɧɚɫɬɭɩɥɟɧɢɹ ɫɨɛɵɬɢɹ ȼ ɡɚɜɢɫɢɬ ɨɬ ɬɨɝɨ ɩɪɨɢɡɨɲɥɨ ɢɥɢ ɧɟɬ ɫɨɛɵɬɢɟ Ⱥ.
ȼ ɬɟɨɪɢɢ ɜɟɪɨɹɬɧɨɫɬɟɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ ɫɜɹɡɢ ɫɨɛɵɬɢɣ Ⱥ ɢ ȼ ɫɥɭɠɢɬ ɬɚɤ ɧɚɡɵɜɚɟɦɚɹ ɭɫɥɨɜɧɚɹ ɜɟɪɨɹɬɧɨɫɬɶ P(B / A) . ȼɟɥɢɱɢɧɚ
P(B / A) ɦɨɠɟɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶɫɹ ɤɚɤ ɜɟɪɨɹɬɧɨɫɬɶ ɧɚɫɬɭɩɥɟɧɢɹ ɫɨɛɵɬɢɹ
ȼ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɫɨɛɵɬɢɟ Ⱥ ɭɠɟ ɧɚɫɬɭɩɢɥɨ.
ɉɪɨɫɬɟɣɲɢɦɢ ɩɪɢɦɟɪɚɦɢ ɫɜɹɡɢ ɫɨɛɵɬɢɣ Ⱥ ɢ ȼ ɦɨɝɭɬ ɫɥɭɠɢɬɶ ɞɜɚ ɤɪɚɣɧɢɯ ɫɥɭɱɚɹ.
1.ɇɚɫɬɭɩɥɟɧɢɟ ɫɨɛɵɬɢɹ Ⱥ ɜɟɞɟɬ ɤ ɨɛɹɡɚɬɟɥɶɧɨɦɭ ɧɚɫɬɭɩɥɟɧɢɸ
ɫɨɛɵɬɢɹ ȼ, ɬ.ɟ. P(B / A) 1.
ɉɪɢɦɟɪ 43. ȼ ɫɭɧɞɭɤɟ ɥɟɠɚɬ ɞɜɟ ɦɨɧɟɬɵ: ɡɨɥɨɬɚɹ ɢ ɫɟɪɟɛɪɹɧɚɹ. ɇɚɭɝɚɞ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɜɵɧɢɦɚɸɬ ɨɛɟ ɦɨɧɟɬɵ. Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɜɬɨɪɚɹ ɦɨɧɟɬɚ ɨɤɚɠɟɬɫɹ ɫɟɪɟɛɪɹɧɨɣ, ɟɫɥɢ ɩɟɪɜɨɣ ɜɵɧɭɥɢ ɡɨɥɨɬɭɸ ɦɨɧɟɬɭ?
ɉɭɫɬɶ ɫɨɛɵɬɢɟ Ⱥ ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɜɵɧɭɥɢ ɡɨɥɨɬɭɸ ɦɨɧɟɬɭ», ɚ ɫɨɛɵɬɢɟ ȼ ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɜɵɧɭɥɢ ɫɟɪɟɛɪɹɧɭɸ ɦɨɧɟɬɭ». ɉɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɡɚɪɚɧɟɟ ɢɡɜɟɫɬɧɨ, ɱɬɨ ɩɟɪɜɨɣ ɜɵɧɭɥɢ ɡɨɥɨɬɭɸ ɦɨɧɟɬɭ (ɫɨɛɵɬɢɟ Ⱥ ɭɠɟ ɩɪɨɢɡɨɲɥɨ) ɢ ɜ ɹɳɢɤɟ ɨɫɬɚɥɚɫɶ ɬɨɥɶɤɨ ɫɟɪɟɛɪɹɧɚɹ ɦɨɧɟɬɚ. ɉɨɷɬɨɦɭ
P(B / A) |
m |
1 |
1 , ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɫɨɛɵɬɢɟ ȼ (ɜɬɨɪɚɹ ɦɨɧɟɬɚ ɨɤɚɠɟɬɫɹ |
|||
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n |
1 |
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ɫɟɪɟɛɪɹɧɨɣ) ɹɜɥɹɟɬɫɹ ɞɨɫɬɨɜɟɪɧɵɦ, ɬ.ɟ. ɬɨɱɧɨ ɩɪɨɢɡɨɣɞɟɬ.
2.ɇɚɫɬɭɩɥɟɧɢɟ ɫɨɛɵɬɢɹ Ⱥ ɢɫɤɥɸɱɚɟɬ ɜɨɡɦɨɠɧɨɫɬɶ ɧɚɫɬɭɩɥɟɧɢɹ
ɫɨɛɵɬɢɹ ȼ, ɬ.ɟ. P(B / A) 0 .
92
ɉɪɢɦɟɪ 44. ȼ ɹɳɢɤɟ ɥɟɠɚɬ ɱɟɬɵɪɟ ɲɚɪɚ: ɬɪɢ ɱɟɪɧɵɯ ɢ ɨɞɢɧ ɛɟɥɵɣ. ɇɚɭɝɚɞ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɜɵɧɢɦɚɸɬ ɞɜɚ ɲɚɪɚ. ɉɟɪɜɵɦ ɜɵɧɭɥɢ ɛɟɥɵɣ ɲɚɪ. Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɜɬɨɪɨɣ ɲɚɪ ɨɤɚɠɟɬɫɹ ɬɨɠɟ ɛɟɥɵɦ?
ɉɨ ɭɫɥɨɜɢɸ ɩɪɢɦɟɪɚ ɡɚɪɚɧɟɟ ɢɡɜɟɫɬɧɨ, ɱɬɨ ɩɟɪɜɵɦ ɜɵɧɭɥɢ ɛɟɥɵɣ ɲɚɪ (ɫɨɛɵɬɢɟ Ⱥ ɭɠɟ ɩɪɨɢɡɨɲɥɨ) ɢ ɜ ɹɳɢɤɟ ɨɫɬɚɥɢɫɶ ɬɪɢ ɱɟɪɧɵɯ ɲɚɪɚ.
ɉɨɷɬɨɦɭ |
P(B / A) |
m |
|
0 |
0 |
, ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɫɨɛɵɬɢɟ ȼ |
(ɜɬɨɪɨɣ ɲɚɪ |
|
n |
3 |
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ɨɤɚɠɟɬɫɹ ɬɨɠɟ ɛɟɥɵɦ) ɹɜɥɹɟɬɫɹ ɧɟɜɨɡɦɨɠɧɵɦ, ɬ.ɟ. ɬɨɱɧɨ ɧɟ ɩɪɨɢɡɨɣɞɟɬ.
Ɍɟɨɪɟɦɚ ɭɦɧɨɠɟɧɢɟ ɜɟɪɨɹɬɧɨɫɬɟɣ. ȼɟɪɨɹɬɧɨɫɬɶ ɫɨɜɦɟɫɬɧɨɝɨ ɧɚ-
ɫɬɭɩɥɟɧɢɹ ɞɜɭɯ ɫɨɛɵɬɢɣ ɪɚɜɧɚ ɩɪɨɢɡɜɟɞɟɧɢɸ ɨɞɧɨɝɨ ɢɡ ɧɢɯ ɧɚ ɭɫɥɨɜɧɭɸ ɜɟɪɨɹɬɧɨɫɬɶ ɞɪɭɝɨɝɨ, ɜɵɱɢɫɥɟɧɧɭɸ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ, ɱɬɨ ɩɟɪɜɨɟ ɫɨɛɵɬɢɟ ɭɠɟ ɧɚɫɬɭɩɢɥɨ:
P(A B) P(A) P(B / A) P(B) P(A / B) .
ɉɪɢɦɟɪ 45. ɂɡ 30 ɷɤɡɚɦɟɧɚɰɢɨɧɧɵɯ ɡɚɞɚɱ ɫɬɭɞɟɧɬ ɭɦɟɟɬ ɪɟɲɚɬɶ ɬɨɥɶɤɨ 25. ɇɚ ɷɤɡɚɦɟɧɟ ɫɬɭɞɟɧɬ ɧɚɭɝɚɞ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɜɵɛɢɪɚɟɬ ɞɜɟ ɡɚɞɚɱɢ. Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɩɟɪɜɭɸ ɡɚɞɚɱɭ ɨɧ ɧɟ ɪɟɲɢɬ, ɚ ɫɨ ɜɬɨɪɨɣ ɫɩɪɚɜɢɬɫɹ?
ɉɭɫɬɶ ɫɨɛɵɬɢɟ Ⱥ ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɫɬɭɞɟɧɬ ɧɟ ɪɟɲɢɬ ɩɟɪɜɭɸ ɡɚɞɚɱɭ», ɬ.ɟ. ɟɦɭ ɩɨɩɚɞɟɬɫɹ ɨɞɧɚ ɢɡ ɩɹɬɢ «ɩɥɨɯɢɯ» ɡɚɞɚɱ. Ɍɨɝɞɚ ɢɫɯɨɞɹ ɢɡ ɤɥɚɫɫɢɱɟɫɤɨɝɨ ɨɩɪɟɞɟɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ ɢɦɟɟɦ:
P(A) |
m |
|
5 |
|
1 |
. |
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n |
30 |
6 |
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ɋɨɛɵɬɢɟ ȼ ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɫɬɭɞɟɧɬ ɪɟɲɢɬ ɜɬɨɪɭɸ ɡɚɞɚɱɭ». ɉɨɫɤɨɥɶɤɭ ɩɨɫɥɟ ɧɚɫɬɭɩɥɟɧɢɹ ɫɨɛɵɬɢɹ Ⱥ ɨɞɧɚ ɢɡ «ɩɥɨɯɢɯ» ɡɚɞɚɱ ɭɠɟ ɢɡɜɥɟɱɟɧɚ, ɬɨ ɨɫɬɚɟɬɫɹ ɜɫɟɝɨ 29 ɡɚɞɚɱ, ɢɡ ɤɨɬɨɪɵɯ 25 ɫɬɭɞɟɧɬ ɭɦɟɟɬ ɪɟɲɚɬɶ. Ɍɨɝɞɚ ɢɫɯɨɞɹ ɢɡ ɤɥɚɫɫɢɱɟɫɤɨɝɨ ɨɩɪɟɞɟɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ ɢɦɟɟɦ:
P(B / A) |
m |
|
25 |
. |
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n |
29 |
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ɂɫɤɨɦɭɸ ɜɟɪɨɹɬɧɨɫɬɶ ɧɚɯɨɞɢɦ ɩɨ ɬɟɨɪɟɦɟ ɭɦɧɨɠɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ:
P(A B) P(A) P(B / A) |
1 |
|
25 |
| 0,14 . |
|
6 |
29 |
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Ɍɟɨɪɟɦɚ ɭɦɧɨɠɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ ɦɨɠɟɬ ɛɵɬɶ ɨɛɨɛɳɟɧɚ ɧɚ ɥɸɛɨɟ ɱɢɫɥɨ ɦɧɨɠɢɬɟɥɟɣ, ɧɚɩɪɢɦɟɪ ɞɥɹ ɬɪɟɯ ɫɨɛɵɬɢɣ ɨɧɚ ɢɦɟɟɬ ɜɢɞ:
93
P(A B C) P(A) P(B / A) P(C / AB) .
ɉɪɢɦɟɪ 46. ɂɡ ɤɨɥɨɞɵ ɜ 52 ɤɚɪɬɵ ɜɵɧɢɦɚɸɬ ɧɚɭɝɚɞ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɬɪɢ ɤɚɪɬɵ. Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɩɟɪɜɨɣ ɤɚɪɬɨɣ ɛɭɞɟɬ ɬɪɨɣɤɚ, ɜɬɨɪɨɣ ɫɟɦɟɪɤɚ, ɚ ɬɪɟɬɶɟɣ ɬɭɡ?
ɉɭɫɬɶ ɫɨɛɵɬɢɟ Ⱥ ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɩɟɪɜɚɹ ɜɵɧɭɬɚɹ ɤɚɪɬɚ ɬɪɨɣɤɚ». Ɍɚɤ ɤɚɤ ɜɫɟɯ ɤɚɪɬ 52 ɢ ɬɪɨɟɤ ɢɡ ɧɢɯ ɱɟɬɵɪɟ, ɬɨ ɢɫɯɨɞɹ ɢɡ ɤɥɚɫɫɢɱɟɫɤɨɝɨ ɨɩɪɟɞɟɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ ɢɦɟɟɦ:
P(A) |
m |
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4 |
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1 |
. |
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n |
52 |
13 |
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ɋɨɛɵɬɢɟ ȼ ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɜɬɨɪɚɹ ɜɵɧɭɬɚɹ ɤɚɪɬɚ ɫɟɦɟɪɤɚ». ɉɨɫɥɟ ɧɚɫɬɭɩɥɟɧɢɹ ɫɨɛɵɬɢɹ Ⱥ (ɜɵɧɭɥɢ ɬɪɨɣɤɭ) ɜ ɤɨɥɨɞɟ ɨɫɬɚɧɟɬɫɹ 51 ɤɚɪɬɚ. Ɍɚɤ ɤɚɤ ɜɫɟɯ ɤɚɪɬ 51 ɢ ɫɟɦɟɪɨɤ ɢɡ ɧɢɯ ɱɟɬɵɪɟ, ɬɨ ɢɫɯɨɞɹ ɢɡ ɤɥɚɫɫɢɱɟɫɤɨɝɨ ɨɩɪɟɞɟɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ ɢɦɟɟɦ:
P(B / A) |
m |
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4 |
. |
n |
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51 |
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ɋɨɛɵɬɢɟ ɋ ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɬɪɟɬɶɹ ɜɵɧɭɬɚɹ ɤɚɪɬɚ ɬɭɡ». ɉɨɫɥɟ ɧɚɫɬɭɩɥɟɧɢɹ ɫɨɛɵɬɢɹ Ⱥ (ɜɵɧɭɥɢ ɬɪɨɣɤɭ) ɢ ɫɨɛɵɬɢɹ ȼ (ɜɵɧɭɥɢ ɫɟɦɟɪɤɭ) ɜ ɤɨɥɨɞɟ ɨɫɬɚɧɟɬɫɹ 50 ɤɚɪɬ. Ɍɚɤ ɤɚɤ ɜɫɟɯ ɤɚɪɬ 50 ɢ ɬɭɡɨɜ ɢɡ ɧɢɯ ɱɟɬɵɪɟ, ɬɨ ɢɫɯɨɞɹ ɢɡ ɤɥɚɫɫɢɱɟɫɤɨɝɨ ɨɩɪɟɞɟɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ ɢɦɟɟɦ:
P(C / AB) |
m |
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4 |
. |
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n |
50 |
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ɋɨɛɵɬɢɟ «ɜɵɧɭɥɢ ɬɪɨɣɤɭ ɢ ɫɟɦɟɪɤɭ ɢ ɬɭɡ» ɨɡɧɚɱɚɟɬ ɫɨɜɦɟɫɬɧɨɟ ɧɚɫɬɭɩɥɟɧɢɟ ɫɨɛɵɬɢɣ Ⱥ, ȼ, ɋ, ɩɨɷɬɨɦɭ ɢɫɤɨɦɭɸ ɜɟɪɨɹɬɧɨɫɬɶ ɧɚɯɨɞɢɦ ɩɨ ɬɟɨɪɟɦɟ ɭɦɧɨɠɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ:
P(A B C) P(A) P(B / A) P(C / AB) |
|
1 |
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4 |
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4 |
0,00048 . |
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13 |
51 |
50 |
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ɉɪɢɦɟɪ 47. ɂɡ ɲɟɫɬɢ ɢɦɟɸɳɢɯɫɹ ɤɥɸɱɟɣ ɤ ɞɜɟɪɢ ɩɨɞɯɨɞɢɬ ɬɨɥɶɤɨ ɞɜɚ. ɋɥɭɱɚɣɧɵɦ ɨɛɪɚɡɨɦ (ɛɟɡ ɜɨɡɜɪɚɳɟɧɢɹ) ɜɵɛɢɪɚɸɬ ɤɥɸɱ ɢ ɩɪɨɛɭɸɬ ɨɬɤɪɵɬɶ ɞɜɟɪɶ. Ʉɥɸɱɢ ɜɵɛɢɪɚɸɬɫɹ ɞɨ ɬɟɯ ɩɨɪ, ɩɨɤɚ ɧɟ ɩɨɹɜɢɬɫɹ ɧɭɠɧɵɣ ɤɥɸɱ. Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ, ɱɬɨ ɤ ɞɜɟɪɢ ɩɨɞɨɣɞɟɬ ɬɨɥɶɤɨ ɱɟɬɜɟɪɬɵɣ ɤɥɸɱ?
ɉɭɫɬɶ ɫɨɛɵɬɢɟ Ⱥ ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɩɟɪɜɵɣ ɤɥɸɱ ɧɟ ɩɨɞɯɨɞɢɬ». Ɍɚɤ ɤɚɤ ɜɫɟɝɨ ɤɥɸɱɟɣ ɲɟɫɬɶ, ɱɟɬɵɪɟ ɢɡ ɤɨɬɨɪɵɯ ɧɟ ɩɨɞɯɨɞɹɬ, ɬɨ ɢɫɯɨɞɹ ɢɡ ɤɥɚɫɫɢɱɟɫɤɨɝɨ ɨɩɪɟɞɟɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ ɢɦɟɟɦ:
P(A) |
m |
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4 |
. |
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n |
6 |
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ɋɨɛɵɬɢɟ ȼ ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɜɬɨɪɨɣ ɤɥɸɱ ɧɟ ɩɨɞɯɨɞɢɬ». Ɍɚɤ ɤɚɤ ɩɨɫɥɟ ɧɚɫɬɭɩɥɟɧɢɹ ɫɨɛɵɬɢɹ Ⱥ ɨɫɬɚɧɟɬɫɹ ɜɫɟɝɨ ɩɹɬɶ ɤɥɸɱɟɣ, ɢɡ ɤɨɬɨɪɵɯ
94
ɬɪɢ ɧɟ ɩɨɞɯɨɞɹɬ, ɬɨ ɢɫɯɨɞɹ ɢɡ ɤɥɚɫɫɢɱɟɫɤɨɝɨ ɨɩɪɟɞɟɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ ɢɦɟɟɦ:
P(B / A) |
m |
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3 |
. |
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n |
5 |
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ɋɨɛɵɬɢɟ ɋ ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɬɪɟɬɢɣ ɤɥɸɱ ɧɟ ɩɨɞɯɨɞɢɬ». ɍɱɢɬɵɜɚɹ, ɱɬɨ ɫɨɛɵɬɢɹ Ⱥ ɢ ȼ ɭɠɟ ɧɚɫɬɭɩɢɥɢ, ɢ ɢɫɯɨɞɹ ɢɡ ɤɥɚɫɫɢɱɟɫɤɨɝɨ ɨɩɪɟɞɟɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ ɢɦɟɟɦ:
P(C / AB) |
m |
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2 |
. |
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n |
4 |
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ɋɨɛɵɬɢɟ D ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɱɟɬɜɟɪɬɵɣ ɤɥɸɱ ɩɨɞɯɨɞɢɬ». ɍɱɢɬɵɜɚɹ, ɱɬɨ ɫɨɛɵɬɢɹ Ⱥ ɢ ȼ ɢ ɋ ɭɠɟ ɧɚɫɬɭɩɢɥɢ ɢ ɢɫɯɨɞɹ ɢɡ ɤɥɚɫɫɢɱɟɫɤɨɝɨ ɨɩɪɟɞɟɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ ɢɦɟɟɦ:
P(D / ABC) |
m |
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2 |
. |
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n |
3 |
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ɋɨɛɵɬɢɟ «ɩɟɪɜɵɣ ɤɥɸɱ ɧɟ ɩɨɞɯɨɞɢɬ ɢ ɜɬɨɪɨɣ ɤɥɸɱ ɧɟ ɩɨɞɯɨɞɢɬ ɢ ɬɪɟɬɢɣ ɤɥɸɱ ɧɟ ɩɨɞɯɨɞɢɬ ɢ ɱɟɬɜɟɪɬɵɣ ɤɥɸɱ ɩɨɞɯɨɞɢɬ» ɨɡɧɚɱɚɟɬ ɫɨɜɦɟɫɬɧɨɟ ɧɚɫɬɭɩɥɟɧɢɟ ɫɨɛɵɬɢɣ Ⱥ, ȼ, ɋ, D, ɩɨɷɬɨɦɭ ɢɫɤɨɦɭɸ ɜɟɪɨɹɬɧɨɫɬɶ ɧɚɯɨɞɢɦ ɩɨ ɬɟɨɪɟɦɟ ɭɦɧɨɠɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ:
P(A B C D) P(A) P(B / A) P(C / AB) P(D / ABC) |
4 |
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3 |
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2 |
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2 |
| 0,13 . |
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6 |
5 |
4 |
3 |
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ɋɨɛɵɬɢɟ ȼ ɧɚɡɵɜɚɟɬɫɹ ɧɟɡɚɜɢɫɢɦɵɦ ɨɬ ɫɨɛɵɬɢɹ Ⱥ, ɟɫɥɢ ɜɟɪɨɹɬɧɨɫɬɶ ɧɚɫɬɭɩɥɟɧɢɹ ɫɨɛɵɬɢɹ ȼ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɬɨɝɨ ɩɪɨɢɡɨɲɥɨ ɢɥɢ ɧɟɬ ɫɨɛɵɬɢɟ Ⱥ.
ɇɟɡɚɜɢɫɢɦɨɫɬɶ ɫɨɛɵɬɢɣ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɧɚɫɬɭɩɥɟɧɢɟ ɫɨɛɵɬɢɹ Ⱥ ɧɟ ɢɡɦɟɧɹɟɬ ɜɟɪɨɹɬɧɨɫɬɢ ɩɨɹɜɥɟɧɢɹ ɫɨɛɵɬɢɹ ȼ, ɬ.ɟ. ɭɫɥɨɜɧɚɹ ɜɟɪɨɹɬɧɨɫɬɶ ɪɚɜɧɚ ɛɟɡɭɫɥɨɜɧɨɣ: P(B / A) P(B) ɢ ɬɟɨɪɟɦɚ ɭɦɧɨɠɟɧɢɹ ɜɟɪɨɹɬɧɨ-
ɫɬɟɣ ɢɦɟɟɬ ɭɩɪɨɳɟɧɧɵɣ ɜɢɞ:
P(A B) P(A) P(B) .
Ⱦɥɹ ɧɟɡɚɜɢɫɢɦɵɯ ɫɨɛɵɬɢɣ ɬɟɨɪɟɦɚ ɭɦɧɨɠɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ ɦɨɠɟɬ ɛɵɬɶ ɬɚɤɠɟ ɨɛɨɛɳɟɧɚ ɧɚ ɥɸɛɨɟ ɱɢɫɥɨ ɦɧɨɠɢɬɟɥɟɣ, ɧɚɩɪɢɦɟɪ, ɞɥɹ ɬɪɟɯ ɫɨɛɵɬɢɣ ɨɧɚ ɢɦɟɟɬ ɜɢɞ:
P(A B C) P(A) P(B) P(C) .
ɉɪɢɦɟɪ 48. Ⱥɛɨɧɟɧɬ ɡɚɛɵɥ ɩɨɫɥɟɞɧɢɟ ɬɪɢ ɰɢɮɪɵ ɬɟɥɟɮɨɧɧɨɝɨ ɧɨɦɟɪɚ, ɧɨ ɩɨɦɧɢɬ, ɱɬɨ ɜɫɟ ɨɧɢ ɧɟɱɟɬɧɵɟ. Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɚɛɨɧɟɧɬ ɞɨɡɜɨɧɢɬɫɹ ɫ ɩɟɪɜɨɝɨ ɪɚɡɚ?
95
ɉɭɫɬɶ ɫɨɛɵɬɢɹ Ⱥ, ȼ ɢ ɋ ɡɚɤɥɸɱɚɸɬɫɹ ɜ ɬɨɦ, ɱɬɨ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɩɟɪɜɚɹ, ɜɬɨɪɚɹ ɢ ɬɪɟɬɶɹ ɢɡ ɡɚɛɵɬɵɯ ɰɢɮɪ ɛɭɞɭɬ ɧɚɛɪɚɧɵ ɜɟɪɧɨ. ɂɫɯɨɞɹ ɢɡ ɬɨɝɨ, ɱɬɨ ɫɨɛɵɬɢɹ Ⱥ, ȼ, ɋ ɧɟɡɚɜɢɫɢɦɵ, ɚ ɬɚɤɠɟ ɭɱɢɬɵɜɚɹ, ɱɬɨ ɢɡ ɩɹɬɢ ɧɟɱɟɬɧɵɯ ɰɢɮɪ ɩɨɞɯɨɞɢɬ ɜ ɤɚɠɞɨɦ ɫɥɭɱɚɟ ɬɨɥɶɤɨ ɨɞɧɚ (ɤɥɚɫɫɢɱɟɫɤɨɟ ɨɩɪɟɞɟɥɟɧɢɟ), ɜɵɱɢɫɥɢɦ ɢɫɤɨɦɭɸ ɜɟɪɨɹɬɧɨɫɬɶ:
P(A B C) P(A) P(B) P(C) |
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1 |
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1 |
0,008 . |
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5 |
5 |
5 |
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ɉɪɢɦɟɪ 49. ɂɦɟɟɬɫɹ ɧɟɢɫɩɪɚɜɧɵɣ ɩɪɢɛɨɪ. ɋɨɛɵɬɢɟ Ⱥ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɜ ɧɟɢɫɩɪɚɜɧɨɦ ɩɪɢɛɨɪɟ «ɩɟɪɟɝɨɪɟɥ ɩɪɟɞɨɯɪɚɧɢɬɟɥɶ». ɋɨɛɵɬɢɟ ȼ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɜ ɧɟɢɫɩɪɚɜɧɨɦ ɩɪɢɛɨɪɟ «ɩɪɨɢɡɨɲɟɥ ɨɛɪɵɜ ɩɪɨɜɨɞɚ». ȼɵɹɫɧɢɦ, ɡɚɜɢɫɢɦɵ ɢɥɢ ɧɟɬ ɫɨɛɵɬɢɹ Ⱥ ɢ ȼ, ɟɫɥɢ ɢɡɜɟɫɬɧɨ, ɱɬɨ:
P(A B) 0, 4 , P(A / B) |
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4 |
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7 |
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Ɍɚɤ ɤɚɤ |
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P(A) P(B / A) P(B) P(A / B) , |
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P(A B) |
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ɬɨ ɢɦɟɟɦ |
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P(A B) |
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P(A) |
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0, 4 |
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7 |
0,7 ; |
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P(B / A) |
4 |
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P(B) |
P(A B) |
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0, 4 |
3 |
0,6 . |
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P(A / B) |
2 |
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Ɍɨɝɞɚ ɢɡ P(A B) |
0, 4 ɢ P(A) P(B) |
0,7 0,6 |
0, 42 ɫɥɟɞɭɟɬ, ɱɬɨ |
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P(A B) ζ P(A) P(B) . ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɫɨɛɵɬɢɹ Ⱥ ɢ ȼ ɹɜɥɹɸɬɫɹ ɡɚɜɢɫɢɦɵɦɢ.
ɉɪɢɦɟɪ 50. Ʉɨɥɟɫɨ ɮɨɪɬɭɧɵ ɪɚɡɞɟɥɟɧɨ ɧɚ 3 ɪɚɜɧɵɯ ɫɟɤɬɨɪɚ, ɩɪɨɧɭɦɟɪɨɜɚɧɧɵɯ ɰɢɮɪɚɦɢ 1, 2, 3. Ʉɨɥɟɫɨ ɤɪɭɬɹɬ ɞɜɚ ɪɚɡɚ. ɋɨɛɵɬɢɟ Ⱥ ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɜ ɩɟɪɜɵɣ ɪɚɡ ɜɵɩɚɥɚ ɰɢɮɪɚ 3». ɋɨɛɵɬɢɟ ȼ ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɜɨ ɜɬɨɪɨɣ ɪɚɡ ɜɵɩɚɥɚ ɰɢɮɪɚ ɧɟ ɛɨɥɶɲɟ ɱɟɦ ɜ ɩɟɪɜɵɣ». ɇɚɣɞɟɦ P(A) , P(B) , P(A B) , P(A / B) ɢ ɨɩɪɟɞɟɥɢɦ, ɹɜɥɹɸɬɫɹ ɥɢ ɧɟɡɚɜɢɫɢ-
ɦɵɦɢ ɫɨɛɵɬɢɹ Ⱥ ɢ ȼ.
Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɭɤɚɡɚɧɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɮɨɪɦɭɥɨɣ ɤɥɚɫɫɢɱɟɫɤɨɝɨ ɨɩɪɟɞɟɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ. Ɉɩɵɬ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɬɨɦ, ɱɬɨ «ɤɨɥɟɫɨ ɤɪɭɬɹɬ ɞɜɚ ɪɚɡɚ». ȼɫɟ ɜɨɡɦɨɠɧɵɟ ɢɫɯɨɞɵ ɨɩɵɬɚ: {11, 12, 13, 21, 22, 23, 31, 32, 33}.
1. ɋɨɛɵɬɢɟ Ⱥ ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɜ ɩɟɪɜɵɣ ɪɚɡ ɜɵɩɚɥɚ ɰɢɮɪɚ 3». ɂɫɯɨɞɵ, ɛɥɚɝɨɩɪɢɹɬɧɵɟ ɫɨɛɵɬɢɸ Ⱥ: {31, 32, 33}:
P(A) |
m |
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3 |
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1 |
. |
n |
9 |
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3 |
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96
2. ɋɨɛɵɬɢɟ ȼ ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɜɨ ɜɬɨɪɨɣ ɪɚɡ ɜɵɩɚɥɚ ɰɢɮɪɚ ɧɟ ɛɨɥɶɲɟ ɱɟɦ ɜ ɩɟɪɜɵɣ».
ɂɫɯɨɞɵ, ɛɥɚɝɨɩɪɢɹɬɧɵɟ ɫɨɛɵɬɢɸ ȼ: {11, 21, 22, 31, 32, 33}:
P(B) |
m |
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6 |
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2 |
. |
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n |
9 |
3 |
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3. ɋɨɛɵɬɢɟ A B ɨɡɧɚɱɚɟɬ ɫɨɜɦɟɫɬɧɨɟ ɧɚɫɬɭɩɥɟɧɢɟ ɫɨɛɵɬɢɣ Ⱥ ɢ ȼ. Ɍɨ ɟɫɬɶ «ɜ ɩɟɪɜɵɣ ɪɚɡ ɜɵɩɚɥɚ ɰɢɮɪɚ 3» ɢ «ɜɨ ɜɬɨɪɨɣ ɪɚɡ ɜɵɩɚɥɚ ɰɢɮɪɚ ɧɟ ɛɨɥɶɲɟ ɱɟɦ ɜ ɩɟɪɜɵɣ».
ɂɫɯɨɞɵ, ɛɥɚɝɨɩɪɢɹɬɧɵɟ ɫɨɛɵɬɢɸ A B : {31, 32, 33}:
P(A B) |
m |
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3 |
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1 |
. |
n |
9 |
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3 |
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4. ɋɨɛɵɬɢɟ A / B ɨɡɧɚɱɚɟɬ ɧɚɫɬɭɩɥɟɧɢɟ ɫɨɛɵɬɢɹ Ⱥ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɫɨɛɵɬɢɟ ȼ ɭɠɟ ɧɚɫɬɭɩɢɥɨ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɫɨɛɵɬɢɟ ȼ ɫɬɚɧɨɜɢɬɫɹ ɞɨɫɬɨɜɟɪɧɵɦ. ȼ ɪɚɦɤɚɯ ɤɥɚɫɫɢɱɟɫɤɨɝɨ ɨɩɪɟɞɟɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ ɭɫɥɨɜɧɭɸ ɜɟɪɨɹɬɧɨɫɬɶ ɦɨɠɧɨ ɜɵɱɢɫɥɹɬɶ ɤɚɤ ɨɬɧɨɲɟɧɢɟ ɱɢɫɥɚ ɢɫɯɨɞɨɜ, ɛɥɚɝɨɩɪɢɹɬɧɵɯ ɩɨɹɜɥɟɧɢɸ ɫɨɛɵɬɢɹ A B (ɫɨɜɦɟɫɬɧɨɦɭ ɩɨɹɜɥɟɧɢɸ ɫɨɛɵɬɢɣ Ⱥ ɢ ȼ), ɤ ɱɢɫɥɭ ɢɫɯɨɞɨɜ, ɛɥɚɝɨɩɪɢɹɬɧɵɯ ɩɨɹɜɥɟɧɢɸ ɫɨɛɵɬɢɹ ȼ (ɨɬɞɟɥɶɧɨɦɭ ɩɨɹɜɥɟɧɢɸ ɫɨɛɵɬɢɹ ȼ)
P(A / B) |
P(A B) |
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1/ 3 |
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3 |
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1 |
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P(B) |
2 / 3 |
6 |
2 |
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5.ɋɨɛɵɬɢɹ Ⱥ ɢ ȼ ɧɟ ɹɜɥɹɸɬɫɹ ɧɟɡɚɜɢɫɢɦɵɦɢ, ɬɚɤ ɤɚɤ
P(A B) |
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1 |
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, P(A) P(B) |
1 |
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2 |
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2 |
, |
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3 |
3 |
3 |
9 |
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ɫɥɟɞɨɜɚɬɟɥɶɧɨ |
P(A B) ζ P(A) P(B) . |
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Ɍɟɨɪɟɦɚ ɫɥɨɠɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ ɧɟɫɨɜɦɟɫɬɧɵɯ ɫɨɛɵɬɢɣ. ȼɟɪɨ-
ɹɬɧɨɫɬɶ ɫɭɦɦɵ ɞɜɭɯ ɧɟɫɨɜɦɟɫɬɧɵɯ ɫɨɛɵɬɢɣ Ⱥ ɢ ȼ ɪɚɜɧɚ ɫɭɦɦɟ ɢɯ ɜɟɪɨɹɬɧɨɫɬɟɣ:
P(A B) P(A) P(B) .
ɉɪɢɦɟɪ 51. ɂɡ ɤɨɥɨɞɵ ɜ 36 ɤɚɪɬ ɧɚɭɝɚɞ ɜɵɧɢɦɚɸɬ ɨɞɧɭ ɤɚɪɬɭ. Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɷɬɨ ɛɭɞɟɬ ɫɟɦɟɪɤɚ ɢɥɢ ɤɨɪɨɥɶ?
ɉɭɫɬɶ ɫɨɛɵɬɢɟ Ⱥ ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɜɵɧɭɬɚɹ ɤɚɪɬɚ ɫɟɦɟɪɤɚ», ɚ ɫɨɛɵɬɢɟ ȼ, ɱɬɨ «ɜɵɧɭɬɚɹ ɤɚɪɬɚ ɤɨɪɨɥɶ». ɂɫɩɨɥɶɡɭɹ ɤɥɚɫɫɢɱɟɫɤɨɟ ɨɩɪɟɞɟɥɟɧɢɟ, ɧɚɯɨɞɢɦ ɜɟɪɨɹɬɧɨɫɬɢ ɷɬɢɯ ɫɨɛɵɬɢɣ:
P(A) |
m |
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4 |
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1 |
, ɬɚɤ ɤɚɤ ɜɫɟɯ ɤɚɪɬ 36, ɚ ɫɟɦɟɪɨɤ ɢɡ ɧɢɯ 4; |
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36 |
9 |
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P(B) |
m |
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4 |
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1 |
, ɬɚɤ ɤɚɤ ɜɫɟɯ ɤɚɪɬ 36, ɚ ɤɨɪɨɥɟɣ ɢɡ ɧɢɯ 4. |
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36 |
9 |
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97 |
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Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɫɨɛɵɬɢɹ Ⱥ ɢ ȼ ɧɟɫɨɜɦɟɫɬɧɵ. Ɍɨɝɞɚ, ɩɪɢɦɟɧɹɹ ɬɟɨɪɟɦɭ ɫɥɨɠɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ ɧɟɫɨɜɦɟɫɬɧɵɯ ɫɨɛɵɬɢɣ, ɩɨɥɭɱɚɟɦ:
P(A B) P(A) P(B) |
1 |
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1 |
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2 |
| 0, 22 . |
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9 |
9 |
9 |
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Ɍɟɨɪɟɦɚ ɫɥɨɠɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ ɧɟɫɨɜɦɟɫɬɧɵɯ ɫɨɛɵɬɢɣ ɦɨɠɟɬ ɛɵɬɶ ɨɛɨɛɳɟɧɚ ɧɚ ɥɸɛɨɟ ɱɢɫɥɨ ɫɥɚɝɚɟɦɵɯ:
P(A1 A2 ... An ) P(A1) P(A2 ) ... P(An ) ,
ɝɞɟ A1, A2, ..., An — ɩɨɩɚɪɧɨ ɧɟɫɨɜɦɟɫɬɧɵ.
ɉɪɢɦɟɪ 52. ɇɚ ɡɚɨɱɧɨɦ ɨɬɞɟɥɟɧɢɢ ɢɧɫɬɢɬɭɬɚ ɨɛɭɱɚɸɬɫɹ 36 ɱɟɥɨɜɟɤ. ɂɡ ɧɢɯ 8 ɱɟɥɨɜɟɤ ɧɚ ɸɪɢɞɢɱɟɫɤɨɦ ɮɚɤɭɥɶɬɟɬɟ, 12 ɱɟɥɨɜɟɤ ɧɚ ɷɤɨɧɨɦɢɱɟɫɤɨɦ ɮɚɤɭɥɶɬɟɬɟ, 10 ɱɟɥɨɜɟɤ ɧɚ ɮɚɤɭɥɶɬɟɬɟ ɩɨɞɝɨɬɨɜɤɢ ɩɫɢɯɨɥɨɝɨɜ ɢ 6 ɱɟɥɨɜɟɤ ɧɚ ɥɢɧɝɜɢɫɬɢɱɟɫɤɨɦ ɮɚɤɭɥɶɬɟɬɟ. Ɉɩɪɟɞɟɥɢɬɶ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɧɚɭɝɚɞ ɜɵɛɪɚɧɧɵɣ ɫɬɭɞɟɧɬ ɭɱɢɬɫɹ ɧɚ ɸɪɢɞɢɱɟɫɤɨɦ, ɷɤɨɧɨɦɢɱɟɫɤɨɦ ɢɥɢ ɥɢɧɝɜɢɫɬɢɱɟɫɤɨɦ ɮɚɤɭɥɶɬɟɬɟ.
ɉɭɫɬɶ ɫɨɛɵɬɢɟ Ⱥ ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɫɬɭɞɟɧɬ ɭɱɢɬɫɹ ɧɚ ɸɪɢɞɢɱɟɫɤɨɦ ɮɚɤɭɥɶɬɟɬɟ», ɫɨɛɵɬɢɟ ȼ, ɱɬɨ «ɫɬɭɞɟɧɬ ɭɱɢɬɫɹ ɧɚ ɷɤɨɧɨɦɢɱɟɫɤɨɦ ɮɚɤɭɥɶɬɟɬɟ», ɚ ɫɨɛɵɬɢɟ ɋ, ɱɬɨ «ɫɬɭɞɟɧɬ ɭɱɢɬɫɹ ɧɚ ɚɝɪɚɪɧɨɦ ɮɚɤɭɥɶɬɟɬɟ». ɂɫɩɨɥɶɡɭɹ ɤɥɚɫɫɢɱɟɫɤɨɟ ɨɩɪɟɞɟɥɟɧɢɟ, ɧɚɯɨɞɢɦ ɜɟɪɨɹɬɧɨɫɬɢ ɷɬɢɯ ɫɨɛɵɬɢɣ:
P(A) |
m |
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8 |
, ɬɚɤ ɤɚɤ ɜɫɟɯ ɫɬɭɞɟɧɬɨɜ 36, ɚ ɸɪɢɫɬɨɜ ɢɡ ɧɢɯ 8; |
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n |
36 |
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P(B) |
m |
12 |
, ɬɚɤɤɚɤɜɫɟɯɫɬɭɞɟɧɬɨɜ36, ɚ ɷɤɨɧɨɦɢɫɬɨɜɢɡɧɢɯ12; |
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36 |
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P(ɋ) |
m |
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6 |
, ɬɚɤ ɤɚɤ ɜɫɟɯ ɫɬɭɞɟɧɬɨɜ 36, ɚ ɥɢɧɝɜɢɫɬɨɜ ɢɡ ɧɢɯ 6. |
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36 |
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Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɫɨɛɵɬɢɹ Ⱥ, ȼ ɢ ɋ ɩɨɩɚɪɧɨ ɧɟɫɨɜɦɟɫɬɧɵ. Ɍɨɝɞɚ, ɩɪɢɦɟɧɹɹ ɬɟɨɪɟɦɭ ɫɥɨɠɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ ɧɟɫɨɜɦɟɫɬɧɵɯ ɫɨɛɵɬɢɣ, ɩɨɥɭɱɚɟɦ:
P(A B C) P(A) P(B) P(C) |
8 |
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12 |
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6 |
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26 |
| 0,72 . |
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36 |
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ȼ ɱɚɫɬɧɨɫɬɢ, ɟɫɥɢ ɫɨɛɵɬɢɹ |
A1, A2, ..., An |
ɨɛɪɚɡɭɸɬ ɩɨɥɧɭɸ ɝɪɭɩɩɭ |
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ɩɨɩɚɪɧɨ ɧɟɫɨɜɦɟɫɬɧɵɯ ɫɨɛɵɬɢɣ, ɬɨ: |
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P(A1 A2 ... An ) |
P(A1) P(A2 ) ... P(An ) |
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ɉɪɢɦɟɪ 53. ȼ ɹɳɢɤɟ 6 ɤɪɚɫɧɵɯ ɲɚɪɨɜ, 8 ɫɢɧɢɯ ɲɚɪɨɜ, 13 ɛɟɥɵɯ ɲɚɪɨɜ ɢ 7 ɱɟɪɧɵɯ ɲɚɪɨɜ. Ɉɩɪɟɞɟɥɢɦ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɧɚɭɝɚɞ ɜɵɧɭɬɵɣ ɨɞɢɧ ɲɚɪ ɛɭɞɟɬ ɱɟɪɧɨɝɨ ɢɥɢ ɛɟɥɨɝɨ ɢɥɢ ɤɪɚɫɧɨɝɨ ɢɥɢ ɫɢɧɟɝɨ ɰɜɟɬɚ.
98
ɉɭɫɬɶ ɫɨɛɵɬɢɹ A1, A2, A3, A4 ɨɡɧɚɱɚɸɬ, ɱɬɨ ɜɵɧɭɬ ɤɪɚɫɧɵɣ, ɫɢɧɢɣ, ɛɟɥɵɣ, ɱɟɪɧɵɣ ɲɚɪ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. ɂɫɩɨɥɶɡɭɹ ɤɥɚɫɫɢɱɟɫɤɨɟ ɨɩ-
ɪɟɞɟɥɟɧɢɟ, |
ɧɚɯɨɞɢɦ |
ɜɟɪɨɹɬɧɨɫɬɢ |
ɷɬɢɯ |
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ɫɨɛɵɬɢɣ: |
P(A1) |
6 |
, |
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P(A2) |
8 |
, |
P(A3) |
13 |
, P(A4) |
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7 |
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Ɉɱɟɜɢɞɧɨ, ɱɬɨ A1, A2, A3, A4 |
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ɨɛɪɚɡɭɸɬ ɩɨɥɧɭɸ ɝɪɭɩɩɭ ɩɨɩɚɪɧɨ ɧɟɫɨɜɦɟɫɬɧɵɯ ɫɨɛɵɬɢɣ, ɬɨ: |
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P(A1 A2 A3 A4) |
P(A1) P(A2 ) P(A3 ) P(A4) |
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Ɍɚɤ ɨɱɟɜɢɞɧɨ, ɱɬɨ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɟ ɫɨɛɵɬɢɹ A ɢ |
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ɧɟɫɨɜɦɟɫɬɧɵ |
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A |
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ɢ ɨɛɪɚɡɭɸɬ ɩɨɥɧɭɸ ɝɪɭɩɩɭ, ɩɨɷɬɨɦɭ P(A |
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P(A) P( |
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1 , ɨɬɤɭɞɚ |
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ɩɨɥɭɱɚɟɦ ɮɨɪɦɭɥɭ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɝɨ ɫɨɛɵɬɢɹ:
P(A) 1 P(A) .
ɉɪɢɦɟɪ 54. ȼɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɧɢ ɨɞɢɧ ɢɡ ɬɪɟɯ ɷɤɡɚɦɟɧɨɜ ɧɟ ɛɭɞɟɬ ɫɞɚɧ ɪɚɜɧɚ 0,1. ɇɚɣɞɟɦ ɜɟɪɨɹɬɧɨɫɬɶ ɫɞɚɱɢ ɯɨɬɹ ɛɵ ɨɞɧɨɝɨ ɢɡ ɬɪɟɯ ɷɤɡɚɦɟɧɨɜ.
ɉɭɫɬɶ ɫɨɛɵɬɢɟ Ⱥ ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɧɢ ɨɞɢɧ ɢɡ ɬɪɟɯ ɷɤɡɚɦɟɧɨɜ ɧɟ ɛɭ-
ɞɟɬ ɫɞɚɧ», ɬɨɝɞɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɟ ɫɨɛɵɬɢɟ A ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɛɭɞɟɬ ɫɞɚɧ ɯɨɬɹ ɛɵ ɨɞɢɧ ɢɡ ɬɪɟɯ ɷɤɡɚɦɟɧɨɜ». ɂɫɩɨɥɶɡɭɹ ɮɨɪɦɭɥɭ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɝɨ ɫɨɛɵɬɢɹ, ɧɚɯɨɞɢɦ:
P(A) 1 P(A) 1 0,1 0,9 .
Ɍɟɨɪɟɦɚ ɫɥɨɠɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ ɫɨɜɦɟɫɬɧɵɯ ɫɨɛɵɬɢɣ. ȼɟɪɨɹɬ-
ɧɨɫɬɶ ɫɭɦɦɵ ɞɜɭɯ ɫɨɜɦɟɫɬɧɵɯ ɫɨɛɵɬɢɣ Ⱥ ɢ ȼ ɪɚɜɧɚ ɫɭɦɦɟ ɜɟɪɨɹɬɧɨɫɬɟɣ ɷɬɢɯ ɫɨɛɵɬɢɣ ɛɟɡ ɜɟɪɨɹɬɧɨɫɬɢ ɢɯ ɩɪɨɢɡɜɟɞɟɧɢɹ:
P(A B) P(A) P(B) P(A B) .
ɉɪɢɦɟɪ 55. ɗɥɟɤɬɪɢɱɟɫɤɚɹ ɰɟɩɶ ɫɨɫɬɨɢɬ ɢɡ ɞɜɭɯ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɫɨɟɞɢɧɟɧɧɵɯ ɷɥɟɦɟɧɬɨɜ. ȼɟɪɨɹɬɧɨɫɬɶ ɨɬɤɚɡɚ ɩɟɪɜɨɝɨ ɷɥɟɦɟɧɬɚ ɪɚɜɧɚ 0,1, ɜɬɨɪɨɝɨ ɪɚɜɧɚ 0,4. ɇɚɣɞɟɦ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɡɚ ɝɨɞ ɨɬɤɚɠɟɬ ɯɨɬɹ ɛɵ ɨɞɢɧ ɷɥɟɦɟɧɬ.
ɉɭɫɬɶ ɫɨɛɵɬɢɟ Ⱥ ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɨɬɤɚɠɟɬ ɩɟɪɜɵɣ ɷɥɟɦɟɧɬ», ɚ ɫɨɛɵɬɢɟ ȼ ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɨɬɤɚɠɟɬ ɜɬɨɪɨɣ ɷɥɟɦɟɧɬ». ɋɨɛɵɬɢɟ «ɨɬɤɚɠɟɬ ɯɨɬɹ ɛɵ ɨɞɢɧ ɷɥɟɦɟɧɬ» ɨɡɧɚɱɚɟɬ, ɱɬɨ ɢɥɢ ɨɬɤɚɠɟɬ ɩɟɪɜɵɣ ɷɥɟɦɟɧɬ ɢɥɢ ɨɬɤɚɠɟɬ ɜɬɨɪɨɣ ɷɥɟɦɟɧɬ ɢɥɢ ɷɬɢ ɞɜɚ ɫɨɛɵɬɢɹ ɧɚɫɬɭɩɹɬ ɨɞɧɨɜɪɟɦɟɧɧɨ
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(ɩɟɪɜɵɣ ɷɥɟɦɟɧɬ ɢ ɜɬɨɪɨɣ ɷɥɟɦɟɧɬ ɨɬɤɚɠɭɬ ɜɦɟɫɬɟ) . ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɬɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɫɭɦɦɭ ɞɜɭɯ ɫɨɜɦɟɫɬɧɵɯ ɫɨɛɵɬɢɣ Ⱥ ɢ ȼ:
P(A B) P(A) P(B) P(A B) 0,1 0, 4 0,1 0, 4 0, 46 .
Ɍɟɨɪɟɦɚ ɫɥɨɠɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ ɫɨɜɦɟɫɬɧɵɯ ɫɨɛɵɬɢɣ ɦɨɠɟɬ ɛɵɬɶ ɨɛɨɛɳɟɧɚ ɧɚ ɥɸɛɨɟ ɱɢɫɥɨ ɫɥɚɝɚɟɦɵɯ, ɧɚɩɪɢɦɟɪ, ɞɥɹ ɬɪɟɯ ɫɨɛɵɬɢɣ:
P(A B C)
P(A) P(B) P(C) P(A B) P(A C) P(B C) P(A B C).
ɉɪɢ ɭɜɟɥɢɱɟɧɢɢ ɱɢɫɥɚ ɫɥɚɝɚɟɦɵɯ ɮɨɪɦɭɥɚ ɜɟɪɨɹɬɧɨɫɬɢ ɫɭɦɦɵ ɫɨɜɦɟɫɬɧɵɯ ɫɨɛɵɬɢɣ ɜɫɟ ɛɨɥɟɟ ɭɫɥɨɠɧɹɟɬɫɹ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɰɟɥɟɫɨɨɛɪɚɡɟɧ ɩɟɪɟɯɨɞ ɤ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɦɭ ɫɨɛɵɬɢɸ. ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ
P(A1 A2 ... An ) 1 P(A1) P(A2 ) ... P(An ) .
ɉɪɢɦɟɪ 56. ȼ ɛɚɧɤɟ ɭɫɬɚɧɨɜɥɟɧɵ ɬɪɢ ɫɢɫɬɟɦɵ ɡɚɳɢɬɵ ɨɬ ɨɝɪɚɛɥɟɧɢɹ. ȼɟɪɨɹɬɧɨɫɬɶ, ɱɬɨ ɩɪɢ ɨɝɪɚɛɥɟɧɢɢ ɫɪɚɛɨɬɚɟɬ ɩɟɪɜɚɹ, ɜɬɨɪɚɹ ɢ ɬɪɟɬɶɹ ɫɢɫɬɟɦɵ ɪɚɜɧɚ 0,9, 0,8 ɢ 0,7 ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. ɇɚɣɞɟɦ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɩɪɢ ɨɝɪɚɛɥɟɧɢɢ ɫɪɚɛɨɬɚɟɬ ɯɨɬɹ ɛɵ ɨɞɧɚ ɫɢɫɬɟɦɚ ɡɚɳɢɬɵ.
ɉɭɫɬɶ ɫɨɛɵɬɢɟ Ⱥ ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɩɪɢ ɨɝɪɚɛɥɟɧɢɢ ɫɪɚɛɨɬɚɟɬ ɯɨɬɹ ɛɵ
ɨɞɧɚ ɫɢɫɬɟɦɚ ɡɚɳɢɬɵ». Ɍɨɝɞɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɟ ɫɨɛɵɬɢɟ A ɨɡɧɚɱɚɟɬ, ɱɬɨ «ɧɢ ɨɞɧɚ ɫɢɫɬɟɦɚ ɡɚɳɢɬɵ ɧɟ ɫɪɚɛɨɬɚɟɬ», ɬ.ɟ. «ɧɟ ɫɪɚɛɨɬɚɟɬ ɩɟɪɜɚɹ ɫɢɫɬɟɦɚ ɡɚɳɢɬɵ ɢ ɧɟ ɫɪɚɛɨɬɚɟɬ ɜɬɨɪɚɹ ɫɢɫɬɟɦɚ ɡɚɳɢɬɵ ɢ ɧɟ ɫɪɚɛɨɬɚɟɬ ɬɪɟɬɶɹ ɫɢɫɬɟɦɚ ɡɚɳɢɬɵ». ɉɭɫɬɶ ɫɨɛɵɬɢɹ B, C, D ɨɡɧɚɱɚɸɬ, ɱɬɨ ɫɪɚɛɨɬɚɟɬ ɩɟɪɜɚɹ, ɜɬɨɪɚɹ, ɬɪɟɬɶɹ ɫɢɫɬɟɦɵ ɡɚɳɢɬɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. Ɍɨɝɞɚ
ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɟ ɫɨɛɵɬɢɹ B, C, D ɨɡɧɚɱɚɸɬ, ɱɬɨ ɧɟ ɫɪɚɛɨɬɚɟɬ ɩɟɪɜɚɹ, ɜɬɨɪɚɹ, ɬɪɟɬɶɹ ɫɢɫɬɟɦɵ ɡɚɳɢɬɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. ȼɟɪɨɹɬɧɨɫɬɢ ɫɨ-
ɛɵɬɢɣ B, C, D ɢɡɜɟɫɬɧɵ ɢɡ ɭɫɥɨɜɢɹ ɡɚɞɚɱɢ: P(B) 0,9 , |
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ɂɫɩɨɥɶɡɭɹ ɮɨɪɦɭɥɭ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ ɩɪɨɬɢɜɨɩɨɥɨɠ- |
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ɧɨɝɨ |
ɫɨɛɵɬɢɹ, ɧɚɯɨɞɢɦ: P( |
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ɉɨɫɤɨɥɶɤɭ ɫɨɛɵɬɢɹ B, C, D ɧɟɡɚɜɢɫɢɦɵ, ɬɨ
P(A) P(B) P(C) P(D) .
ɉɨɷɬɨɦɭɢɫɤɨɦɚɹɜɟɪɨɹɬɧɨɫɬɶ (ɜɟɪɨɹɬɧɨɫɬɶɫɨɛɵɬɢɹȺ) ɛɭɞɟɬɪɚɜɧɚ:
P(A) P(B C D) 1 P(A) 1 P(B) P(C) P(D) 1 0,1 0, 2 0,3 0,994.
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