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Консультация № 180575

05.11.2010, 05:31

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Образование

Здравствуйте, уважаемые эксперты!
Помогите пожалуйста с решением следующей задачи:
Игральную кость подбрасывают 100 раз. Найти вероятнейшее число выпадений «6» и вероятность такого результата.

Обсуждение

давно

Орловский Дмитрий

Мастер-Эксперт
319965
1463

05.11.2010, 10:39

общий

это ответ

Здравствуйте, Litta!
Имеем схему Бернулли с вероятностью успеха p=1/6. Наивероятнейшее число k находится из неравенства
np-q[$8804$]k<np+p (q=1-p)
В нашем случае (p=1/6, q=5/6, n=100) имеем неравенство 15,83[$8804$]k<16,83 ---> k=16

Вероятность этого события находим по формуле Бернулли
p_n(k)=C₁₀₀¹⁶p^k(1-p)^n-k=C₁₀₀¹⁶(1/6)¹⁶(5/6)⁸⁴

Приближенно это значение можно вычислить, пользуясь локальной теоремой Муавра-Лапласа:
p(16)=exp(-0,5x²)/[$8730$](2*pi*npq), где x=(k-np)/[$8730$](npq)
Выполняя вычисления, находим x=-0,18 и p(16)=0,1053

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