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Здравствуйте! У меня возникли сложности с таким вопросом: (Теория Вероятностей)
Вопрос касается задачи, которую легко найти в учебнике по ТВиМС автора Гмурмана(1979 год). Задача номер 183. Приведу условие задачи: Вероятность выигрыша по одному лотерейному билету р=0,01. Сколько нужно купить лотерейных билетов, чтобы выиграть хотя бы по одному из них с вероятностью не меньшей, чем 0,95?
В учебнике приведено решение этой задачи. Ход представленного решения мне понятен. Я не могу понять, почему решать нужно именно таким образом. Я рассуждаю так: если купить 1 билет, то вероятность выигрыша будет 1 процент (согласно условию задачи). Какова вероятность удачи, при покупке мной двух билетов? Выигрыш по первому билету и выигрыш по второму - события несовместные. Тогда по теореме о сложении вероятностей несовместных событий, вероятность выигрыша при купленных мной двух билетах: Р(2_билета_куплено)=Р(выиграет первый)+Р(выиграет_второй)=2*р=0,02 По понятной "логике", вероятность выигрыша, требуемая в условии, будет достигнута при покупке мной 95 билетов(правильный ответ: >300).
Я последовательно прорешиваю почти все задачи из этого учебника, поэтому если объснение моего вопроса может быть проиллюстрировано примером из этого же учебника(или какой-либо задачей вообще, которая бы показала мне, в чем я путаюсь), то я был бы вдвойне благодарен.
"Выигрыш по первому и выигрыш по второму - события несовместимые" -- это неверно. Такое может быть, если билетов всего 100, и из них только один выигрышный. Вероятность выигрыша для первого билета p = 0.01, а для последующих вероятность выигрыша возрастает. В этом случае ваши рассуждения верны, но это не та задача, которую требуется решить.
В лотерее билетов обычно гораздо больше нескольких сотен, и вероятность выигрыша p для первого и последующих билетов можно считать одинаковой. В этом случае вероятность Pn(k) выигрыша ровно k билетов из n подчиняется биномиальному распределению. Считая данную в задаче вероятность p = 0.01 малой, можно воспользоваться распределением Пуассона, которое аппроксимирует биномиальное распределение при p -> 0. Это решение приведено в учебнике, ответ n > 300.
Задачу можно решить проще и точнее. Требуется определить количество билетов n, при котором с вероятностью не менее 0.95 выиграет хотя бы один. Это то же самое, что мы допускаем, что ни один из n билетов не выиграет с вероятностью не более 1 - 0.95 = 0.05. Вероятность того, что выбранный билет не выигрывает, равна 1-p = 0.99. Вероятность того, что ни один из n билетов не выигрывает, равна (1-p)n = 0.99n (произведение вероятностей независимых событий). Эта вероятность должна быть меньше 0.05. Решая уравнение 0.99n = 0.05 логарифмированием правой и левой частей, получим n > 298.
Я бы решал эту задачу так. Пусть - вероятность того, что купленный билет выигрышный. Тогда - вероятность того, что купленный билет не выигрышный. Предположим, что куплено билетов. Вероятность того, что все они окажутся невыигрышными, равна Вероятность того, что среди купленных билетов окажется хотя бы один выигрышный (это противоположное событие), равна По условию задачи она не меньше числа Решим соответствующее неравенство:
Следовательно, нужно купить не менее билетов.
5
Об авторе:
Facta loquuntur.
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