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Консультация № 191634
31.10.2017, 20:13
0.00 руб.
31.10.2017, 20:23
0
2
1
Здравствуйте! Прошу помощи в следующем вопросе:
На сторонах квадрата выбираются случайным образом 3 точки. Найдите вероятность того, что центр квадрата находится внутри треугольника, построенного по выбранным точкам.
На сторонах квадрата выбираются случайным образом 3 точки. Найдите вероятность того, что центр квадрата находится внутри треугольника, построенного по выбранным точкам.
Обсуждение
31.10.2017, 20:24
общий
Обратите, пожалуйста, внимание на данную консультацию, перенесённую из другого раздела.
Об авторе:
Facta loquuntur.
Facta loquuntur.
03.11.2017, 09:14
общий
это ответ
Здравствуйте, Асмик Гаряка!
Точки могут распределиться по сторонам квадрата 64-мя способами (4х4х4=64).
Из них:
i) на одной стороне - 4 способа;
ii) на соседних сторонах - 24 способа;
iii) на противоположных сторонах - 12 способов;
iv) на разных сторонах - 24 способа.
В случаях (i) и (ii) треугольника просто нет или центр квадрата не попадает внутрь.
В случае (iii) вероятность попадания 1/3 - такая же, как вероятность того, что из 3-х точек, брошенных на отрезок, одна из них, заранее выбранная, попадет между двумя другими (см. рисунок).
В случае (iv) после того, как выбраны точки на противоположных сторонах квадрата, исход зависит только от того, какая из двух оставшихся сторон выбрана. Вероятность попадания 1/2.
С учетом числа способов размещения точек на сторонах квадрата получим искомую вероятность:
(12/64)*(1/3) + (24/64)*(1/2) = 1/4.
Ответ: 1/4
Точки могут распределиться по сторонам квадрата 64-мя способами (4х4х4=64).
Из них:
i) на одной стороне - 4 способа;
ii) на соседних сторонах - 24 способа;
iii) на противоположных сторонах - 12 способов;
iv) на разных сторонах - 24 способа.
В случаях (i) и (ii) треугольника просто нет или центр квадрата не попадает внутрь.
В случае (iii) вероятность попадания 1/3 - такая же, как вероятность того, что из 3-х точек, брошенных на отрезок, одна из них, заранее выбранная, попадет между двумя другими (см. рисунок).
В случае (iv) после того, как выбраны точки на противоположных сторонах квадрата, исход зависит только от того, какая из двух оставшихся сторон выбрана. Вероятность попадания 1/2.
С учетом числа способов размещения точек на сторонах квадрата получим искомую вероятность:
(12/64)*(1/3) + (24/64)*(1/2) = 1/4.
Ответ: 1/4
5
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