Введение в теорию вероятностей. Учебное пособие

ɉɭɫɬɶ ɫɪɟɞɢ N ɨɛɴɟɤɬɨɜ ɢɦɟɟɬɫɹ M ɦɟɱɟɧɵɯ ɧɚɩɪɢɦɟɪ ɨɤɪɚɲɟɧɧɵɯM N Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɱɬɨ ɫɪɟɞɢ ɢɡɜɥɟɱɟɧɧɵɯ ɫɥɭɱɚɣɧɵɦ ɨɛɪɚɡɨɦ Ʉ ɨɛɴɟɤɬɨɜ ɨɤɚɠɟɬɫɹ ɪɨɜɧɨ L ɦɟɱɟɧɵɯ d L d M "

Ɋɟɲɟɧɢɟ Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ Ⱥ ɫɨɛɵɬɢɟ ɜɟɪɨɹɬɧɨɫɬɶ ɤɨɬɨɪɨɝɨ ɦɵ ɢɳɟɦ ȼɫɟɜɨɡɦɨɠɧɵɟ ɜɵɛɨɪɤɢ ɨɛɴɟɦɚ Ʉ ɨɛɪɚɡɭɸɬ ɩɪɨɫɬɪɚɧɫɬɜɨ ɷɥɟɦɟɧɬɚɪɧɵɯ ɢɫɯɨ ɞɨɜ ɨɧɢ ɩɨɩɚɪɧɨ ɧɟɫɨɜɦɟɫɬɧɵ ɪɚɜɧɨɜɨɡɦɨɠɧɵ ɢ ɢɯ ɝɪɭɩɩɚ ɩɨɥɧɚ Ɋɚɡɥɢ ɱɚɹɫɶ ɥɢɲɶ ɨɬɨɛɪɚɧɧɵɦɢ ɨɛɴɟɤɬɚɦɢ ɢɫɯɨɞɵ ɹɜɥɹɸɬɫɹ ɫɨɱɟɬɚɧɢɹɦɢ ɬɚɤ ɱɬɨ ɜɨɡɦɨɠɧɨɟ ɢɯ ɱɢɫɥɨ ɟɫɬɶ n CNK Ȼɥɚɝɨɩɪɢɹɬɧɵɣ ɢɫɯɨɞ ɧɚɫɬɭɩɚɟɬ ɤɨɝɞɚ ɨɞɧɨ ɜɪɟɦɟɧɧɨ ɢɡɜɥɟɱɟɧɨ ɪɨɜɧɨ L ɨɛɴɟɤɬɨɜ ɢɡ M ɨɫɬɚɥɶɧɵɟ K L ɢɡɜɥɟɱɟɧɵ ɢɡ N M ɧɟɦɟɱɟɧɵɯ

ɑɢɫɥɨ ɜɨɡɦɨɠɧɵɯ ɢɡɜɥɟɱɟɧɢɣ L ɨɛɴɟɤɬɨɜ ɢɡ M ɟɫɬɶ CML ɬɨɝɞɚ ɤɚɤ ɱɢɫɥɨ ɨɫɬɚɥɶɧɵɯ ɢɡɜɥɟɱɟɧɢɣ ɟɫɬɶ CNK ML ɩɪɢ ɷɬɨɦ ɧɚ ɤɚɠɞɨɟ ɢɡ CML ɢɡɜɥɟɱɟɧɢɣ ɦɟɱɟ ɧɵɯ ɨɛɴɟɤɬɨɜ ɩɪɢɯɨɞɢɬɫɹ ɪɨɜɧɨ CNK ML ɢɡɜɥɟɱɟɧɢɣ ɨɛɴɟɤɬɨɜ ɧɟɦɟɱɟɧɵɯ ɋɨɝɥɚɫɧɨ ɩɪɢɧɰɢɩɭ ɭɦɧɨɠɟɧɢɹ ɤɨɥɢɱɟɫɬɜɨ ɛɥɚɝɨɩɪɢɹɬɧɵɯ ɢɫɯɨɞɨɜ ɟɫɬɶ CML CNK ML ɋɥɟɞɨ ɜɚɬɟɥɶɧɨ

3. Относительная частота и статистическая вероятность

ɉɨɧɹɬɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɱɚɫɬɨɬɵ ɫɨɛɵɬɢɹ ɜɜɨɞɢɬɫɹ ɜ ɪɟɡɭɥɶɬɚɬɟ ɚɧɚɥɢ ɡɚ ɩɪɨɜɟɞɟɧɧɵɯ ɨɩɵɬɨɜ ȿɫɥɢ ɫɨɛɵɬɢɟ A ɩɨɹɜɢɥɨɫɶ m mA ɪɚɡ ɬɨ ɨɬɧɨɫɢɬɟɥɶ ɧɨɣ ɱɚɫɬɨɬɨɣ ɫɨɛɵɬɢɹ A ɧɚɡɵɜɚɸɬ ɱɢɫɥɨ

m W A Wɩ A n

ɉɪɚɤɬɢɤɚ ɩɨɤɚɡɵɜɚɟɬ ɱɬɨ ɫ ɪɨɫɬɨɦ ɱɢɫɥɚ ɨɞɧɨɬɢɩɧɵɯ ɨɩɵɬɨɜ ɨɬɧɨɫɢ ɬɟɥɶɧɚɹ ɱɚɫɬɨɬɚ ɩɪɢɨɛɪɟɬɚɟɬ ɫɜɨɣɫɬɜɨ ɭɫɬɨɣɱɢɜɨɫɬɢ ɤɨɥɟɛɥɹɫɶ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɤɨɬɨɪɨɝɨ ɱɢɫɥɚ Ɋ Ɋ Ⱥ ɤɨɬɨɪɨɟ ɧɚɡɵɜɚɟɬɫɹ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɜɟɪɨɹɬɧɨɫɬɶɸ ɫɨɛɵɬɢɹ A ɇɢɠɟ ɛɭɞɟɬ ɩɪɟɞɥɨɠɟɧɨ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɛɨɫɧɨɜɚɧɢɟ ɷɬɨɝɨ ɫɜɨɣ ɫɬɜɚ ɧɚɡɵɜɚɟɦɨɟ ɡɚɤɨɧɨɦ ɛɨɥɶɲɢɯ ɱɢɫɟɥ Ȼɟɪɧɭɥɥɢ

Ɉɱɟɜɢɞɧɨ ɱɬɨ ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɱɚɫɬɨɬɚ W A ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɫɜɨɣɫɬɜɚɦ ɚɧɚɥɨɝɢɱɧɵɦ ɜ ɱɚɫɬɧɨɫɬɢ dW A d

4. Геометрическая вероятность

Ɋɚɫɫɦɨɬɪɢɦ ɫɥɟɞɭɸɳɢɣ ɨɩɵɬ ɜ ɧɟɤɨɬɨɪɭɸ ɨɝɪɚɧɢɱɟɧɧɨɣ ɨɛɥɚɫɬɢ ȿ ɫɥɭɱɚɣɧɵɦ ɨɛɪɚɡɨɦ ɛɥɭɠɞɚɟɬ ɬɨɱɤɚ ɩɪɢɱɟɦ ɩɨɩɚɞɚɧɢɹ ɟɟ ɜ ɥɸɛɵɟ ɱɚɫɬɢ

21

ɢɦɟɸɳɢɟ ɨɞɢɧɚɤɨɜɭɸ ɦɟɪɭ ɧɚɩɪɢɦɟɪ ɩɥɨɳɚɞɶ ɜ ɫɥɭɱɚɟ ɩɥɨɫɤɢɯ ɨɛɥɚɫɬɟɣ ɫɱɢɬɚɸɬɫɹ ɪɚɜɧɨɜɨɡɦɨɠɧɵɦɢ ɉɭɫɬɶ ɫɨɛɵɬɢɟ Ⱥ ± ɟɟ ɩɨɩɚɞɚɧɢɟ ɜ ɨɛɥɚɫɬɶ A ȿ ȿɫɥɢ S A ɢ S E S E z ± ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɦɟɪɵ Ⱥ ɢ ȿ ɬɨ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɜɟɪɨɹɬɧɨɫɬɶɸ ɫɨɛɵɬɢɹ A ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɱɢɫɥɨ

Ⱦɥɹ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɜɟɪɨɹɬɧɨɫɬɢ ɫɨɯɪɚɧɹɸɬɫɹ ɨɱɟɜɢɞɧɨ ɫɜɨɣɫɬɜɚɢ ɨɰɟɧɤɚ d Ɋ $ d ɞɥɹ ɜɫɹɤɨɝɨ ɫɥɭɱɚɣɧɨɝɨ ɫɨɛɵɬɢɹ Ⱥ Ɂɚɦɟɬɢɦ ɱɬɨ ɡɞɟɫɶ ɪɚɜɟɧɫɬɜɚ P Ⱥ P Ⱥ ɨɤɚɡɵɜɚɸɬɫɹ ɜɨɡɦɨɠɧɵɦɢ ɢ ɞɥɹ ɧɟɤɨɬɨɪɵɯ

ɫɥɭɱɚɣɧɵɯ ɫɨɛɵɬɢɣ Ⱥ Ɍɚɤ ɧɚɩɪɢɦɟɪ ɟɫɥɢ ȿ ± ɤɜɚɞɪɚɬ ɚ ɫɥɭɱɚɣɧɨɟ ɫɨɛɵɬɢɟ Ⱥ ± ɩɨɩɚɞɚɧɢɟ ɬɨɱɤɢ ɧɚ ɥɸɛɭɸ ɟɝɨ ɞɢɚɝɨɧɚɥɶ ɬɨ P Ⱥ ɩɨɫɤɨɥɶɤɭ ɞɢɚɝɨɧɚɥɶ ɢɦɟɟɬ ɧɭɥɟɜɭɸ ɩɥɨɳɚɞɶ

5. Понятие об аксиомах вероятности

Ɋɚɫɫɦɨɬɪɟɧɧɵɟ ɜɵɲɟ ɦɨɞɟɥɢ ɜɟɪɨɹɬɧɨɫɬɢ ɫɨɛɵɬɢɹ ɤɥɚɫɫɢɱɟɫɤɚɹ ɫɬɚ ɬɢɫɬɢɱɟɫɤɚɹ ɝɟɨɦɟɬɪɢɱɟɫɤɚɹ ɨɛɥɚɞɚɸɬ ɧɟɤɨɬɨɪɵɦɢ ɨɛɳɢɦɢ ɫɜɨɣɫɬɜɚɦɢ ɤɨɬɨ ɪɵɟ ɦɨɠɧɨ ɩɨɥɨɠɢɬɶ ɜ ɨɫɧɨɜɭ ɚɤɫɢɨɦɚɬɢɱɟɫɤɨɝɨ ɨɩɪɟɞɟɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ

Ɂɚɞɚɞɢɦ ɜɟɪɨɹɬɧɨɫɬɶ ɤɚɤ ɧɟɤɨɬɨɪɭɸ ɮɭɧɤɰɢɸ Ɋ ɧɚ ɚɥɝɟɛɪɟ ɫɨɛɵɬɢɣ Uɢɥɢ ɞɚɠɟ V ɚɥɝɟɛɪɟ ɫɨɛɵɬɢɣ Ȼɭɞɟɦ ɫɱɢɬɚɬɶ ɱɬɨ ɜɵɩɨɥɧɟɧɵ ɫɥɟɞɭɸɳɢɟ ɫɜɨɣɫɬɜɚ ɚɤɫɢɨɦɵ

Ⱥ ɧɟɨɬɪɢɰɚɬɟɥɶɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ P Ⱥ t ɞɥɹ ɜɫɹɤɨɝɨ A U

Ⱥ ɧɨɪɦɢɪɨɜɚɧɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ Ɋ ȿ

Ⱥ ɚɞɞɢɬɢɜɧɨɟ ɫɜɨɣɫɬɜɨ ɟɫɥɢ ɫɨɛɵɬɢɹ A $ «An ɩɨɩɚɪɧɨ ɧɟɫɨɜ ɦɟɫɬɧɵ ɬɨ

n

P A A … An ¦P Ak

k

Ⱥ ɚɤɫɢɨɦɚ ©ɪɚɫɲɢɪɟɧɧɨɣª ɚɞɞɢɬɢɜɧɨɫɬɢ ɞɥɹ ɩɨɩɚɪɧɨ ɧɟɫɨɜɦɟɫɬɧɵɯ ɫɨɛɵɬɢɣ

f

Ɋ A A An ¦P Ak

k

ɝɞɟ ɱɢɫɥɨɜɨɣ ɪɹɞ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɫɯɨɞɹɳɢɦɫɹ

ɉɪɢ ɜɵɩɨɥɧɟɧɢɢ ɚɤɫɢɨɦ Ⱥ ± Ⱥ ɬɨ ɢɦɟɸɬ ɦɟɫɬɨ ɫɥɟɞɭɸɳɢɟ ɭɬɜɟɪ ɠɞɟɧɢɹ

22

Теорема 5.1. Ⱦɥɹ ɜɫɹɤɨɣ ɩɚɪɵ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɯ ɫɨɛɵɬɢɣ A A ɫɩɪɚ ɜɟɞɥɢɜɨ ɪɚɜɟɧɫɬɜɨ

P A P A

Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ ɜ ɫɢɥɭ ɫɨɨɬɧɨɲɟɧɢɹ A A ȿ ɢ ɫɜɨɣɫɬɜɚ ɧɨɪɦɢɪɨɜɚɧɧɨ ɫɬɢ Ⱥ

Ɋ $ $ Ɋ ȿ Ⱦɚɥɟɟ ɜ ɫɢɥɭ ɧɟɫɨɜɦɟɫɬɧɨɫɬɢ ɫɨɛɵɬɢɣ A A ɢ ɚɤɫɢɨɦɵ Ⱥ ɩɨɥɭɱɚɟɦ

P A P A Ɋ $ $

ɱɬɨ ɢ ɭɬɜɟɪɠɞɚɥɨɫɶ ȼ ɱɚɫɬɧɨɫɬɢ ɞɥɹ ɫɨɛɵɬɢɣ ɞɨɫɬɨɜɟɪɧɨɝɨ ɢ ɧɟɜɨɡɦɨɠɧɨɝɨ ɤɚɤ ɩɪɨɬɢɜɨɩɨ

ɥɨɠɧɵɯ ɩɨɥɭɱɚɟɦ

P Ɋ ȿ

Теорема 5.2. ȼɫɹɤɨɟ ɫɨɛɵɬɢɟ Ⱥ ɢɦɟɟɬ ɜɟɪɨɹɬɧɨɫɬɶ Ɋ $ d Ⱦɥɹ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɪɟɡɭɥɶɬɚɬɨɦ ɬɟɨɪɟɦɵ ɢ ɫɜɨɣɫɬɜɨɦ Ⱥ

P A P A t Ɋ Ⱥ

ȼɵɲɟ ɞɥɹ ɤɥɚɫɫɢɱɟɫɤɨɣ ɜɟɪɨɹɬɧɨɫɬɢ ɭɫɬɚɧɨɜɥɟɧɨ ɱɬɨ ɨɧɚ ɭɞɨɜɥɟɬɜɨ ɪɹɟɬ ɫɜɨɣɫɬɜɚɦ Ⱥ ɢ Ⱥ ɉɪɨɜɟɪɢɦ ɞɥɹ ɧɟɟ ɬɟɩɟɪɶ ɫɩɪɚɜɟɞɥɢɜɨɫɬɶ Ⱥ ɇɚɱɧɟɦ ɫ ɪɚɫɫɦɨɬɪɟɧɢɹ ɞɜɭɯ ɧɟɫɨɜɦɟɫɬɧɵɯ ɫɨɛɵɬɢɣ ɢ ɭɫɬɚɧɨɜɢɦ ɱɬɨ

P A A P A P A

ɉɭɫɬɶ ɨɩɵɬ ɢɦɟɟɬ n ɷɥɟɦɟɧɬɚɪɧɵɯ ɢɫɯɨɞɨɜ ɢɡ ɤɨɬɨɪɵɯ m ɛɥɚɝɨɩɪɢɹɬ ɫɬɜɭɟɬ ɫɨɛɵɬɢɸ A ɚ m ɢɫɯɨɞɨɜ ± A ȼɜɢɞɭ ɧɟɫɨɜɦɟɫɬɧɨɫɬɢ ɫɨɛɵɬɢɣ A ɢ A ɫɪɟɞɢ m ɢ m2 ɧɟɬ ɨɛɳɢɯ ɢɫɯɨɞɨɜ ɋɥɟɞɨɜɚɬɟɥɶɧɨ ɫɨɛɵɬɢɸ A A ɛɥɚɝɨɩɪɢɹɬ ɫɬɜɭɸɬ ɪɨɜɧɨ m m ɢɫɯɨɞɨɜ ɚ ɬɨɝɞɚ

ɍɬɜɟɪɠɞɟɧɢɟ ɩ ɦɨɠɧɨ ɪɚɫɩɪɨɫɬɪɚɧɢɬɶ ɬɟɩɟɪɶ ɧɚ ɩɪɨɢɡɜɨɥɶɧɨɟ ɤɨ ɧɟɱɧɨɟ ɤɨɥɢɱɟɫɬɜɨ ɫɨɛɵɬɢɣ ɩɨɥɶɡɭɹɫɶ ɚɫɫɨɰɢɚɬɢɜɧɵɦ ɫɜɨɣɫɬɜɨɦ ɫɥɨɠɟɧɢɹ ɢ ɞɟɣɫɬɜɭɹ ɩɨ ɢɧɞɭɤɰɢɢ Ɍɚɤ ɧɚɩɪɢɦɟɪ ɟɫɥɢ ɬɪɢ ɫɨɛɵɬɢɹ A A A ɩɨɩɚɪɧɨ ɧɟɫɨɜɦɟɫɬɧɵ ɬɨ ɫɨɛɵɬɢɹ A A ɢ A ɨɱɟɜɢɞɧɨ ɛɭɞɭɬ ɧɟɫɨɜɦɟɫɬɧɵɦɢ Ɍɟɩɟɪɶ ɧɚ ɨɫɧɨɜɚɧɢɢ ɭɬɜɟɪɠɞɟɧɢɹ ɩ ɩɨɥɭɱɚɟɦ

ɬɚɤ ɱɬɨ ɫɜɨɣɫɬɜɨ Ⱥɨɤɚɡɵɜɚɟɬɫɹ ɜɟɪɧɵɦ ɢ ɜ ɫɥɭɱɚɟ ɬɪɟɯ ɫɥɚɝɚɟɦɵɯ

23

6. Вероятность произведения событий

Ɋɚɫɫɦɨɬɪɢɦ ɫɩɨɫɨɛ ɜɵɱɢɫɥɟɧɢɹ ɤɥɚɫɫɢɱɟɫɤɨɣ ɜɟɪɨɹɬɧɨɫɬɢ ɩɪɨɢɡɜɟɞɟ ɧɢɹ ɫɨɛɵɬɢɣ A B Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ PA B ɜɟɪɨɹɬɧɨɫɬɶ ɫɨɛɵɬɢɹ B ɜɵɱɢɫɥɟɧ

ɧɭɸ ɩɪɢ ɭɫɥɨɜɢɢ ɱɬɨ A ɩɪɨɢɡɨɲɥɨ ɢ ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɟɟ ɭɫɥɨɜɧɨɣ ɜɟɪɨɹɬɧɨɫɬɶɸ ɫɨɛɵɬɢɹ ȼ ɜɟɪɨɹɬɧɨɫɬɶɸ ȼ ɩɨ ɧɚɫɬɭɩɥɟɧɢɢ Ⱥ

ɪɨɜɧɨ m ɛɥɚɝɨɩɪɢɹɬɫɬɜɭɸɬ ɩɪɨɢɡɜɟɞɟɧɢɸ A B ɢ mȺ ɢɫɯɨɞɨɜ ɛɥɚɝɨɩɪɢɹɬ ɫɬɜɭɸɬ ɫɨɛɵɬɢɸ A Ɍɨɝɞɚ

Ⱦɥɹ ɷɬɨɝɨ ɡɚɦɟɬɢɦ ɱɬɨ ɩɨ ɧɚɫɬɭɩɥɟɧɢɢ A ɥɢɲɶ ɬɟ mȺ ɢɫɯɨɞɨɜ ɤɨɬɨɪɵɟ ɟɦɭ ɛɥɚɝɨɩɪɢɹɬɫɬɜɨɜɚɥɢ ɨɤɚɡɵɜɚɸɬɫɹ ɜɨɡɦɨɠɧɵɦɢ ɞɥɹ ɞɚɥɶɧɟɣɲɟɝɨ ɪɚɫɫɦɨɬɪɟɧɢɹ ɚ ɫɨɛɵɬɢɟ B ɢɦɟɟɬ ɬɨɝɞɚ ɛɥɚɝɨɩɪɢɹɬɧɵɦɢ ɢɫɯɨɞɵ ɛɥɚɝɨɩɪɢɹɬɧɵɟ ɨɞɧɨɜɪɟɦɟɧ ɧɨ ɢ ɞɥɹ A Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɩɨ ɧɚɫɬɭɩɥɟɧɢɢ A ɨɤɚɡɵɜɚɟɬɫɹ mB m ɢ ɜɵɱɢɫɥɟ ɧɢɟ PA B ɩɪɢɜɨɞɢɬ ɤ ɪɚɜɟɧɫɬɜɭ ɋɨɝɥɚɫɧɨ ɪɚɜɟɧɫɬɜɚɦ ɢ ɩɨɥɭ ɱɚɟɦ ɭɬɜɟɪɠɞɟɧɢɟ

ȼ ɫɥɭɱɚɟ ɩɪɨɢɡɜɟɞɟɧɢɹ ɬɪɟɯ ɢ ɛɨɥɶɲɟɝɨ ɱɢɫɥɚ ɫɨɛɵɬɢɣ ɢɦɟɟɬ ɦɟɫɬɨ ɚɧɚɥɨɝɢɱɧɵɣ ɪɟɡɭɥɶɬɚɬ Ɍɚɤ ɧɚɩɪɢɦɟɪ

P ABɋ 3 $ 3A % ɊȺȼ ɋ

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɩɪɨɜɨɞɢɬɫɹ ɩɨ ɢɧɞɭɤɰɢɢ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɫɜɨɣɫɬɜɚ ɚɫɫɨ ɰɢɚɬɢɜɧɨɫɬɢ ɭɦɧɨɠɟɧɢɹ ɫɨɛɵɬɢɣ

Ɂɚɦɟɱɚɧɢɟ ɋɨɨɬɧɨɲɟɧɢɟ

P AB PA B P A

ɜɵɬɟɤɚɸɳɟɟ ɞɥɹ ɤɥɚɫɫɢɱɟɫɤɨɣ ɜɟɪɨɹɬɧɨɫɬɢ ɩɪɢ Ɋ Ⱥ z ɢɡ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɨɠɟɧɨ ɜ ɨɫɧɨɜɭ ɨɩɪɟɞɟɥɟɧɢɹ ɭɫɥɨɜɧɨɣ ɜɟɪɨɹɬɧɨɫɬɢ ɫɨ ɛɵɬɢɹ ȼ

24

ɋɨɛɵɬɢɹ Ⱥ ɢ ȼ ɧɚɡɵɜɚɸɬɫɹ ɧɟɡɚɜɢɫɢɦɵɦɢ ɟɫɥɢ ɜɟɪɨɹɬɧɨɫɬɶ ɤɚɠɞɨɝɨ ɢɡ ɧɢɯ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɬɨɝɨ ɧɚɫɬɭɩɢɥɨ ɥɢ ɢɥɢ ɧɟ ɧɚɫɬɭɩɢɥɨ ɞɪɭɝɨɟ ɫɨɛɵɬɢɟ ɇɚɩɪɢɦɟɪ ɟɫɥɢ ɫɪɟɞɢ N ɩɪɟɞɦɟɬɨɜ ɛɭɞɟɬ ɪɨɜɧɨ M 0 < M < N ɦɟɱɟɧɵɯ ɨɤɪɚ ɲɟɧɧɵɯ ɛɪɚɤɨɜɚɧɧɵɯ ɢ ɬ ɩ ɢ ɩɪɨɢɡɜɨɞɢɬɫɹ ©ɛɟɡɜɨɡɜɪɚɬɧɚɹª ɜɵɛɨɪɤɚ ɬɨ ɜɟ ɪɨɹɬɧɨɫɬɶ ɢɡɜɥɟɱɟɧɢɹ ɜɬɨɪɨɝɨ ɩɪɟɞɦɟɬɚ ɦɟɱɟɧɵɦ ɫɨɛɵɬɢɟ ȼ ɡɚɜɢɫɢɬ ɨɬ ɬɨɝɨ ɛɵɥ ɥɢ ɦɟɱɟɧɵɦ ɫɨɛɵɬɢɟ Ⱥ ɢɡɜɥɟɱɟɧ ɩɟɪɜɵɣ ɩɪɟɞɦɟɬ Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ

M

ɊȺ ȼ N

ɬɚɤ ɤɚɤ ɩɨ ɧɚɫɬɭɩɥɟɧɢɢ Ⱥ ɫɪɟɞɢ N ɩɪɟɞɦɟɬɨɜ ɨɫɬɚɥɨɫɶ M ɦɟɱɟɧɵɯ ɢ

Ɉɞɧɚɤɨ ɜ ɫɥɭɱɚɟ ɜɨɡɜɪɚɬɧɨɣ ɜɵɛɨɪɤɢ ɜɟɪɨɹɬɧɨɫɬɶ ɢɡɜɥɟɱɟɧɢɹ ɜɬɨɪɨɝɨ ɩɪɟɞɦɟɬɚ ɦɟɱɟɧɵɦ ɛɵɥɚ ɛɵ ɨɞɧɨɣ ɢ ɬɨɣ ɠɟ ɧɟɡɚɜɢɫɢɦɨ ɨɬ ɬɨɝɨ ɤɚɤɢɦ ɛɵɥ ɩɟɪɜɵɣ ɢɡ ɩɪɟɞɦɟɬɨɜ

M

Ɋ ȼ ɊȺ ȼ ɊȺ ȼ N

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɜ ɩɟɪɜɨɦ ɫɥɭɱɚɟ ɫɨɛɵɬɢɹ Ⱥ ɢ ȼ ɡɚɜɢɫɢɦɵ ɚ ɜɨ ɜɬɨɪɨɦ ɫɥɭ ɱɚɟ ± ɧɟɡɚɜɢɫɢɦɵ

Ⱥɧɚɥɨɝɢɱɧɨ ɫɨɛɵɬɢɹ A A Ⱥn ɧɚɡɵɜɚɸɬɫɹ ɧɟɡɚɜɢɫɢɦɵɦɢ ɜ ɫɨɜɨɤɭɩ ɧɨɫɬɢ ɟɫɥɢ ɤɚɠɞɚɹ ɢɡ ɜɟɪɨɹɬɧɨɫɬɟɣ Pj P Aj ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ ɜ ɭɫɥɨɜɢɹɯ

ɞɚɧɧɵɯ ɨɩɵɬɨɜ ɂɡ ɬɟɨɪɟɦɵ ɩɪɢɦɟɧɟɧɧɨɣ ɤ ɞɜɭɦ ɧɟɡɚɜɢɫɢɦɵɦ ɫɨɛɵɬɢɹɦ ɜɵɬɟɤɚɟɬ ɱɬɨ

P AB P A P B

Ɋɚɫɩɪɨɫɬɪɚɧɹɹ ɪɟɡɭɥɶɬɚɬ ɧɚ ɩ ɫɨɛɵɬɢɣ ɧɟɡɚɜɢɫɢɦɵɯ ɜ ɫɨɜɨɤɭɩɧɨɫɬɢ ɩɪɢ ɯɨɞɢɦ ɤ ɫɨɨɬɧɨɲɟɧɢɸ

Ɋ $ $ Ⱥn Ɋ $ Ɋ $ Ɋ Ⱥn

7. Вероятность суммы совместных событий

ȼɵɲɟ ɛɵɥɚ ɪɚɫɫɦɨɬɪɟɧɚ ɜɟɪɨɹɬɧɨɫɬɶ ɫɭɦɦɵ ɩɨɩɚɪɧɨ ɧɟɫɨɜɦɟɫɬɧɵɯ ɫɨɛɵɬɢɣ ɋɥɟɞɭɸɳɢɣ ɪɟɡɭɥɶɬɚɬ ɜ ɫɥɭɱɚɟ ɫɭɦɦɵ ɥɸɛɵɯ ɞɜɭɯ ɫɨɛɵɬɢɣ Ⱥ ɢ ȼ ɹɜ ɥɹɟɬɫɹ ɛɨɥɟɟ ɨɛɳɢɦ

25

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ȼɜɢɞɭ ɩɨɥɧɨɬɵ ɝɪɭɩɩɵ ɝɢɩɨɬɟɡ ɢɦɟɟɦ

ȿɇ ɇ ɇɩ

ɍɱɢɬɵɜɚɹ ɨɱɟɜɢɞɧɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ Ⱥ Ⱥ ȿ ɢ ɫɜɨɣɫɬɜɚ ɨɩɟɪɚɰɢɣ ɫɥɨɠɟ ɧɢɹ ɢ ɭɦɧɨɠɟɧɢɹ ɩɨɥɭɱɚɟɦ ɬɨɝɞɚ

ȺȺ ȿ ɇ Ⱥ ɇ Ⱥ ɇɩ Ⱥ

ɋɨɛɵɬɢɹ ɇk Ⱥ N ɩ ɨɤɚɡɵɜɚɸɬɫɹ ɩɨɩɚɪɧɨ ɧɟɫɨɜɦɟɫɬɧɵɦɢ ɩɪɟɞɩɨ

ɥɨɠɢɜ ɩɪɨɬɢɜɧɨɟ ɦɵ ɩɨɥɭɱɢɥɢ ɛɵ ɫɨɜɦɟɫɬɧɨɫɬɶ ɝɢɩɨɬɟɡ ɚ ɩɨɷɬɨɦɭ

n

P A ¦P Hk A

k

ɑɬɨɛɵ ɭɫɬɚɧɨɜɢɬɶ ɨɫɬɚɥɨɫɶ ɩɪɢɦɟɧɢɬɶ ɤ ɤɚɠɞɨɦɭ ɫɥɚɝɚɟɦɨɦɭ ɩɨ ɫɥɟɞɧɟɣ ɫɭɦɦɵ ɬɟɨɪɟɦɭ ɭɦɧɨɠɟɧɢɹ

ȼ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ ɞɜɭɯ ɝɢɩɨɬɟɡ ɮɨɪɦɭɥɚ ɩɨɥɧɨɣ ɜɟɪɨɹɬɧɨɫɬɢ ɩɪɢɧɢɦɚɟɬ

ɜɢɞ

P A P H PH A P H PH A

ɉɭɫɬɶ ɜɵɩɨɥɧɟɧɵ ɭɫɥɨɜɢɹ ɩ ɢ ɩɪɢ ɷɬɨɦ ɫɨɛɵɬɢɟ Ⱥ ɧɚɫɬɭɩɢɥɨ Ɍɨ ɝɞɚ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ ɱɬɨ ɫɨɛɵɬɢɟ Ⱥ ɨɤɚɡɚɥɨɫɶ ɫɥɟɞɫɬɜɢɟɦ ɝɢɩɨɬɟɡɵ ɢɦɟɧɧɨ ɇk ɨɩɪɟɞɟɥɹɟɬɫɹ ɜ ɜɢɞɟ

ɝɞɟ Ɋ Ⱥ z ± ɩɨɥɧɚɹ ɜɟɪɨɹɬɧɨɫɬɶ ɋɨɨɬɧɨɲɟɧɢɹ ɧɚɡɵɜɚɸɬ ɮɨɪɦɭ ɥɚɦɢ Ȼɟɣɟɫɚ

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɜɵɬɟɤɚɟɬ ɢɡ ɪɚɜɟɧɫɬɜɚ Ⱥ ɇɤ ɇɤ Ⱥ ɩɪɢɦɟɧɢɜ ɤ ɤɨ ɬɨɪɨɦɭ ɬɟɨɪɟɦɭ ɭɦɧɨɠɟɧɢɹ ɩɨɥɭɱɢɦ

P Ⱥ 3Ⱥ +N P Hk PHk A k ɩ

ɱɬɨ ɢ ɪɚɜɧɨɫɢɥɶɧɨ

9. Классическая вероятность: типовые задачи

Задача 9.1. ȼɨ ɜɧɨɜɶ ɩɨɫɬɪɨɟɧɧɨɦ ɞɨɦɟ ɢɦɟɟɬɫɹ ɤɜɚɪɬɢɪ ɫɪɟɞɢ ɤɨ ɬɨɪɵɯ ɪɚɫɩɨɥɨɠɟɧɵ ɧɚ ɤɪɚɣɧɢɯ ɷɬɚɠɚɯ Ʉɜɚɪɬɢɪɵ ɪɚɫɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɠɪɟ ɛɢɸ ɇɚɣɬɢ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ ɱɬɨ ɷɬɢ ɤɜɚɪɬɢɪ ɞɨɫɬɚɧɭɬɫɹ ɞɚɧɧɵɦ ɜɨɫɶɦɢ ɫɟ ɦɶɹɦ ɩɟɪɟɫɟɥɟɧɰɚɦ

27

Ɋɟɲɟɧɢɟ ɉɨɫɤɨɥɶɤɭ ɜɫɟɝɨ ɭɱɚɫɬɧɢɤɨɜ ɠɟɪɟɛɶɟɜɤɢ ± ɬɨ ɷɥɟɦɟɧɬɚɪɧɵɟ ɢɫɯɨɞɵ ɨɩɵɬɚ ± ɷɬɨ ɭɩɨɪɹɞɨɱɟɧɧɵɟ ɜɵɛɨɪɤɢ ɢɡ ɩɨ ɬ ɟ ɩɟɪɟɫɬɚɧɨɜɤɢ Ʉɨɥɢɱɟɫɬɜɨ ɜɫɟɜɨɡɦɨɠɧɵɯ ɢɫɯɨɞɨɜ ɬɚɤɢɦ ɨɛɪɚɡɨɦ ɟɫɬɶ ɩ

ȿɫɥɢ ɫɨɛɵɬɢɟ Ⱥ ɫɨɫɬɨɢɬ ɜ ɬɨɦ ɱɬɨ ɤɜɚɪɬɢɪ ɧɚ ɤɪɚɣɧɢɯ ɷɬɚɠɚɯ ɞɨɫɬɚ ɧɭɬɫɹ ɜɨɫɶɦɢ ɫɟɦɶɹɦ ɩɟɪɟɫɟɥɟɧɰɚɦ ɬɨ ɱɢɫɥɨ ɛɥɚɝɨɩɪɢɹɬɧɵɯ ɢɫɯɨɞɨɜ ɬȺ ɦɨɠɧɨ ɜɵɱɢɫɥɢɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ ɋɪɟɞɢ ɞɚɧɧɵɯ ɜɨɫɶɦɢ ɫɟɦɟɣ ɜɨɡɦɨɠ ɧɨ ɫɩɨɫɨɛɨɜ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɤɜɚɪɬɢɪ ɧɚ ɤɪɚɣɧɢɯ ɷɬɚɠɚɯ ɢ ɤɚɠɞɵɣ ɬɚɤɨɣ ɫɩɨɫɨɛ ɫɨɱɟɬɚɟɬɫɹ ɫ ɨɫɬɚɥɶɧɵɦɢ ɫɩɨɫɨɛɚɦɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɤɜɚɪɬɢɪ ɫɪɟ ɞɢ ɨɫɬɚɥɶɧɵɯ ɭɱɚɫɬɧɢɤɨɜ ɠɟɪɟɛɶɟɜɤɢ ɋɨɝɥɚɫɧɨ ɩɪɢɧɰɢɩɭ ɭɦɧɨɠɟ

Задача 9.2. ȼ ɬɨɪɝɨɜɭɸ ɫɟɬɶ ɩɨɫɬɭɩɚɸɬ ɨɞɧɨɬɢɩɧɵɟ ɬɨɜɚɪɵ ɞɜɭɯ ɩɪɨ ɢɡɜɨɞɢɬɟɥɟɣ ɉɟɪɜɵɣ ɩɪɨɢɡɜɨɞɢɬɟɥɶ ɩɨɫɬɚɜɥɹɟɬ ɫɬɚɧɞɚɪɬɧɵɯ ɢɡɞɟɥɢɣ ɜɬɨɪɨɣ ± Ⱦɥɹ ɤɨɧɬɪɨɥɹ ɜɵɛɪɚɧɨ ɫɥɭɱɚɣɧɵɦ ɨɛɪɚɡɨɦ ɩɨ ɨɞɧɨɦɭ ɢɡɞɟɥɢɸ ɤɚɠɞɨɝɨ ɩɪɨɢɡɜɨɞɢɬɟɥɹ Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɱɬɨ

ɚ ɨɛɚ ɢɡɞɟɥɢɹ ɫɬɚɧɞɚɪɬɧɵ ɛ ɬɨɥɶɤɨ ɨɞɧɨ ɢɡ ɧɢɯ ɫɬɚɧɞɚɪɬɧɨ ɜ ɯɨɬɹ ɛɵ ɨɞɧɨ ɫɬɚɧɞɚɪɬɧɨ

Ɋɟɲɟɧɢɟ ɚ Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ Ⱥ ɫɨɛɵɬɢɟ ɫɨɫɬɨɹɳɟɟ ɜ ɬɨɦ ɱɬɨ ɨɛɚ ɢɡɞɟɥɢɹ ɫɬɚɧɞɚɪɬ

ɧɵ ȼɵɪɚɡɢɦ Ⱥ ɱɟɪɟɡ ɫɨɛɵɬɢɹ Ⱥ ± ɢɡɞɟɥɢɟ ɩɟɪɜɨɝɨ ɩɪɨɢɡɜɨɞɢɬɟɥɹ ɫɬɚɧɞɚɪɬɧɨ ɢ Ⱥ ± ɢɡɞɟɥɢɟ ɜɬɨɪɨɝɨ ɫɬɚɧɞɚɪɬɧɨ əɫɧɨ ɱɬɨ A A A ɩɪɢɱɺɦ ɫɨɛɵɬɢɹ Ⱥ ɢ Ⱥ

ɧɟɡɚɜɢɫɢɦɵ ɋɥɟɞɨɜɚɬɟɥɶɧɨ

P A P A P A

ɛ ɉɭɫɬɶ ɫɨɛɵɬɢɟ ȼ ± ɪɨɜɧɨ ɨɞɧɨ ɢɡɞɟɥɢɟ ɫɬɚɧɞɚɪɬɧɨ ɜ ɷɬɨɦ ɫɥɭɱɚɟ B A A A A ɉɨɫɤɨɥɶɤɭ ɜɟɪɨɹɬɧɨɫɬɢ ɫɨɛɵɬɢɣ Ⱥ ɢ Ⱥ ɩɨɫɬɨɹɧɧɵ ɜ ɭɫɥɨɜɢɹɯ ɩɪɨɜɨɞɢɦɨɝɨ ɤɨɧɬɪɨɥɹ ɬɨ A ɢ A ± ɧɟɡɚɜɢɫɢɦɵ ɬɨɱɧɨ ɬɚɤ ɠɟ ɧɟɡɚɜɢɫɢɦɵ A ɢ A ɢ ɩɪɢ ɷɬɨɦ ɩɪɨɢɡɜɟɞɟɧɢɹ A A ɢ A A ɨɱɟɜɢɞɧɨ ɧɟɫɨɜɦɟɫɬɧɵ Ɍɨɝɞɚ

Задача 9.3. ɉɨ ɫɚɦɨɥɟɬɭ ɜɵɩɭɳɟɧɚ ɪɚɤɟɬɚ ɋɚɦɨɥɟɬ ɦɨɠɟɬ ɭɧɢɱɬɨɠɢɬɶ ɟɟ ɫ ɜɟɪɨɹɬɧɨɫɬɶɸ ȿɫɥɢ ɷɬɨɝɨ ɫɞɟɥɚɬɶ ɧɟ ɭɞɚɟɬɫɹ ɬɨ ɪɚɤɟɬɚ ɩɨɪɚɠɚɟɬ ɫɚɦɨ ɥɟɬ ɫ ɜɟɪɨɹɬɧɨɫɬɶɸ Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɪɚɠɟɧɢɹ ɫɚɦɨɥɟɬɚ ɪɚɤɟɬɨɣ"

Ɋɟɲɟɧɢɟ ɉɨɪɚɠɟɧɢɟ ɫɚɦɨɥɟɬɚ ± ɫɥɨɠɧɨɟ ɫɨɛɵɬɢɟ ɋ ɫɨɫɬɨɹɳɟɟ ɜ ɫɨɜ ɦɟɫɬɧɨɦ ɧɚɫɬɭɩɥɟɧɢɢ ɫɨɛɵɬɢɹ Ⱥ ± ɧɟɭɧɢɱɬɨɠɟɧɢɢ ɪɚɤɟɬɵ ɢ ɫɨɛɵɬɢɹ ȼ ± ɩɨ ɪɚɠɟɧɢɹ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɫɚɦɨɥɟɬɚ ɪɚɤɟɬɨɣ ɋɥɟɞɨɜɚɬɟɥɶɧɨ ɋ Ⱥȼ Ɍɟɩɟɪɶ Ɋ ɋ Ɋ Ⱥ ɊȺ ȼ

ɉɟɪɟɣɞɟɦ ɤ ɧɚɯɨɠɞɟɧɢɸ ɜɟɪɨɹɬɧɨɫɬɟɣ ɋɨɛɵɬɢɟ Ⱥ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ ɫɨɛɵ ɬɢɸ ɭɧɢɱɬɨɠɟɧɢɹ ɪɚɤɟɬɵ ɫɥɟɞɨɜɚɬɟɥɶɧɨ

Ɋ Ⱥ ɩɪɢ ɷɬɨɦ ɊȺ ȼ

Ɉɤɨɧɱɚɬɟɥɶɧɨ

Ɋ ɋ

Задача 9.4. ɢɧɮɨɪɦɚɰɢɢ ɩɨɫɬɭɩɚɟɬ ɫ ɩɟɪɜɨɝɨ ɫɟɪɜɟɪɚ ± ɫɨ ɜɬɨɪɨɝɨ ȼɟɪɨɹɬɧɨɫɬɶ ɩɨɫɬɭɩɥɟɧɢɹ ɮɟɣɤɨɜɨɣ ɢɧɮɨɪɦɚɰɢɢ ɞɥɹ ɩɟɪɜɨɝɨ ɫɟɪɜɟɪɚ ɪɚɜɧɚ Ɋ ɞɥɹ ɜɬɨɪɨɝɨ ± Ɋ

O Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɱɬɨ ɢɧɮɨɪɦɚɰɢɹ ɫ ɧɚɭɝɚɞ ɜɵɛɪɚɧɧɨɝɨ ɫɟɪɜɟɪɚ ɩɨ ɫɬɭɩɢɬ ɮɟɣɤɨɜɚɹ"

ɉɨɫɬɭɩɢɥɚ ɮɟɣɤɨɜɚɹ ɢɧɮɨɪɦɚɰɢɹ Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɱɬɨ ɟɺ ɩɨɫɬɚɜɢɥ ɩɟɪɜɵɣ ɫɟɪɜɟɪ"

Ɋɟɲɟɧɢɟ ɉɭɫɬɶ Ⱥ ± ɫɨɛɵɬɢɟ ɨɛɧɚɪɭɠɟɧɢɹ ɮɟɣɤɚ ȼɨɡɦɨɠɧɵ ɝɢɩɨɬɟɡɵ ɇ ± ɨɧ ɩɨɫɬɭɩɢɥ ɫ ɫɟɪɜɟɪɚ ʋ ɇ ± ɫ ɫɟɪɜɟɪɚ Ɉɱɟɜɢɞɧɨ ɱɬɨ ɜɵɩɨɥɧɟ ɧɵ ɭɫɥɨɜɢɹ ɧɟɫɨɜɦɟɫɬɧɨɫɬɢ ɢ ɩɨɥɧɨɬɵ ɝɪɭɩɩɵ ɇ ɢ ɇ Ɂɞɟɫɶ Ɋ ɇ Ɋ ɇ PH A PH A Ɍɨɝɞɚ

ɩɨ ɮɨɪɦɭɥɟ ɩɨɥɧɨɣ ɜɟɪɨɹɬɧɨɫɬɢ

Задача 9.5. Ʉɨɦɩɶɸɬɟɪɧɨɣ ɞɢɚɝɧɨɫɬɢɤɟ ɩɨɞɜɟɪɝɚɟɬɫɹ ɝɪɭɩɩɚ ɭɱɚɫɬɧɢ ɤɨɜ ɞɢɫɩɚɧɫɟɪɢɡɚɰɢɢ ɫɪɟɞɢ ɤɨɬɨɪɵɯ ɫɬɪɚɞɚɸɬ ɧɟɤɨɬɨɪɵɦɢ ɡɚɛɨɥɟɜɚ ɧɢɹɦɢ

ȼ ɪɟɡɭɥɶɬɚɬɟ ɞɢɚɝɧɨɫɬɢɤɢ ɛɨɥɟɡɧɶ ɜɵɹɜɥɹɟɬɫɹ ɫ ɜɟɪɨɹɬɧɨɫɬɶɸ ɢ ɫ ɜɟ ɪɨɹɬɧɨɫɬɶɸ ɪɚɜɧɨɣ ɡɞɨɪɨɜɵɣ ɭɱɚɫɬɧɢɤ ɩɪɢɡɧɚɟɬɫɹ ɛɨɥɶɧɵɦ ɍ ɩɪɨɢɡɜɨɥɶ ɧɨ ɜɵɛɪɚɧɧɨɝɨ ɩɪɨɬɟɫɬɢɪɨɜɚɧɧɨɝɨ ɭɱɚɫɬɧɢɤɚ ɤɨɦɩɶɸɬɟɪ ɜɵɹɜɢɥ ɡɚɛɨɥɟɜɚɧɢɟ Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɱɬɨ ɩɪɨɢɡɨɲɥɚ ɨɲɢɛɤɚ"

Ɋɟɲɟɧɢɟ ɉɭɫɬɶ ɫɨɛɵɬɢɟ A ɫɨɫɬɨɢɬ ɜ ɬɨɦ ɱɬɨ ɩɪɨɬɟɫɬɢɪɨɜɚɧɧɵɣ ɭɱɚɫɬɧɢɤ ɩɪɢɡɧɚɧ ɛɨɥɶɧɵɦ ȼɨɡɦɨɠɧɵ ɩɪɟɞɩɨɥɨɠɟɧɢɹ ɝɢɩɨɬɟɡɵ

29

H ± ɬɟɫɬɢɪɭɟɬɫɹ ɭɱɚɫɬɧɢɤ ɫɬɪɚɞɚɸɳɢɣ ɡɚɛɨɥɟɜɚɧɢɟɦ

H ± ɬɟɫɬɢɪɭɟɬɫɹ ɡɞɨɪɨɜɵɣ ɭɱɚɫɬɧɢɤ

Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɜɟɪɨɹɬɧɨɫɬɶ ɝɢɩɨɬɟɡɵ H ɩɪɢ ɭɫɥɨɜɢɢ ɱɬɨ ɧɚɫɬɭɩɢɥɨ ɫɨɛɵɬɢɟ Ⱥ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɩɪɢɦɟɧɢɦɚ ɮɨɪɦɭɥɚ Ȼɟɣɟɫɚ

ɉɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɝɢɩɨɬɟɡɵ ɢɦɟɸɬ ɜɟɪɨɹɬɧɨɫɬɢ Ɋ ɇ ɢ Ɋ ɇ ɋɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɫɥɨɜɧɵɟ ɜɟɪɨɹɬɧɨɫɬɢ ɫɨɛɵɬɢɹ ɢɦɟɸɬ ɜɢɞ

Задача 9.6. ɇɚ ɫɤɥɚɞɟ ɜɩɟɪɟɦɟɲɤɭ ɥɟɠɚɬ ɷɤɡɟɦɩɥɹɪɨɜ ɤɧɢɝ ɤɨɬɨɪɵɟ ɹɜɥɹɸɬɫɹ ɩɟɪɜɵɦɢ ɢ ɜɬɨɪɵɦɢ ɬɨɦɚɦɢ ɫɨɱɢɧɟɧɢɣ Ⱦɠ Ɋ Ɋ Ɍɨɥɤɢɟɧɚ ɩɪɢɱɺɦ ɱɢɫɥɨ ɬɟɯ ɢ ɞɪɭɝɢɯ ɛɨɥɶɲɟ ɟɞɢɧɢɰɵ ȼɨɡɦɨɠɧɨ ɥɢ ɩɪɢ ɷɬɨɦ ɭɫɥɨɜɢɢ ɜɡɹɬɶ ɫɥɭ

ɱɚɣɧɵɦ ɨɛɪɚɡɨɦ ɩɚɪɭ ɤɧɢɝ ɱɬɨɛɵ ɫ ɜɟɪɨɹɬɧɨɫɬɶɸ ɪɚɜɧɨɣ ɷɬɨ ɨɤɚɡɚɥɫɹ

ɛɵ ɞɜɭɯɬɨɦɧɢɤ"

Ɋɟɲɟɧɢɟ ɉɭɫɬɶ ɩɟɪɜɵɯ ɬɨɦɨɜ K ɲɬɭɤ ɚ ɜɬɨɪɵɯL ɲɬɭɤ Ɇɨɠɧɨ ɫɱɢɬɚɬɶ ɱɬɨ ɤɧɢɝɢ ɛɟɪɭɬɫɹ ɩɨɨɱɟɪɺɞɧɨ ȼɟɪɨɹɬɧɨɫɬɶ ɜɡɹɬɶ ɫɧɚɱɚɥɚ ɩɟɪɜɵɣ ɬɨɦ ɚ ɡɚɬɟɦ

ɫɮɨɪɦɢɪɨɜɚɬɶ ɞɜɭɯɬɨɦɧɢɤ ɬɟɦ ɢɥɢ ɢɧɵɦ ɫɩɨɫɨɛɨɦ ɩɨ ɮɨɪɦɭɥɟ ɜɟɪɨɹɬɧɨɫɬɢ