Вопрос школьника
Стрелок выполнил 400 выстрелов. Найти вероятность 325 попаданий, если вероятность попадания при каждом выстреле равна 0,8.
Ответ от учителя
Для решения данной задачи необходимо воспользоваться биномиальным распределением.
Биномиальное распределение описывает вероятность того, что в серии из n независимых испытаний с двумя возможными исходами (успехом и неудачей) число успехов равно k.
Формула для биномиального распределения выглядит следующим образом:
P(k) = C(n, k) * p^k * (1-p)^(n-k)
где P(k) — вероятность того, что число успехов равно k, C(n, k) — число сочетаний из n по k, p — вероятность успеха, (1-p) — вероятность неудачи.
В данной задаче n = 400 (количество выстрелов), k = 325 (количество попаданий), p = 0,8 (вероятность попадания при каждом выстреле).
Тогда вероятность того, что стрелок попадет 325 раз из 400, будет равна:
P(325) = C(400, 325) * 0,8^325 * 0,2^75
Вычислим каждый из множителей:
C(400, 325) = 400! / (325! * 75!) = 1,352,078,109,785,600 / (1,080,168,028,761,000 * 2,213,733,925,381) = 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