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Здравствуйте! У меня возникли сложности с таким вопросом: Независимо друг от друга работает несколько топливных насосов. Расчетная вероятность прорыва горючего из трубопровода за время работы насоса составляет Ргор=0,01, вероятность образования окислительной атмосферы Рок=0,055, вероятность возникновения искры Ри=0,008. Сколько насосов могут работать одновременно, чтобы вероятность возникновения пожара не превысила 0,125?
По-видимому, надо полагать, что пожар при работе одного насоса (событие A) возникает в том случае, если произойдёт прорыв горючего и образуется окислительная атмосфера, и возникнет искра. т. е. будут действовать три фактора. Вероятность этого составляет p(A) = pгор [$183$] pок [$183$] pи = 0,01 [$183$] 0,055 [$183$] 0,008 = 4,4 [$183$] 10-6. Вероятность того, что пожар при работе одного насоса не возникнет, составляет p([$172$]A) = 1 - p(A) = 0,9999956.
Если одновременно независимо работает n насосов, то пожар не возникнет (событие Bn) с вероятностью p(Bn) = (p([$172$]A))n, поскольку и первый, и второй, и n-й насосы должны работать без пожара. По условию p(Bn) = 1 - 0,125 [$8805$] 0,875. Тогда (p([$172$]A))n [$8805$] 0,875, (0,9999956)n [$8805$] 0,875, n [$183$] ln 0,9999956 [$8804$] ln 0,875, n [$8804$] ln 0,875 : ln 0,9999956 = 30347,97. Следовательно, вероятность возникновения пожара не превысит 0,125, если одновременно будет работать не более 30347 насосов.
Здравствуйте, Марина! Если я правильно понял физику явления, то пожар возможен в случае прорыва горючего и выполнения одного из двух условий: окислительная атмосфера или искра. В таком случае вероятность пожара на одном насосе Р(А)=Ргор*(Рок+Ри)=0,01*(0,055+0,008)=0,00063 Используя теорему суммирования вероятностей, для k независимых насосов получим, что вероятность возгорания хотя бы на одном равна: Р(А1+...+Аk)=k*Р(А)-Р(А1*...*Аk)=0,00063k-(0,00063)^k=0,00063(k-0,00063^(k-1)) По условию задачи 0,00063(k-0,00063^(k-1))<=0,125 k-0,00063^(k-1)<=198,4 Наибольшее количество насосов - 198.
Второе слагаемое (вычитаемое) всегда меньще единицы и стемится к нулю при возрастании к. Это я могу расписать. Меня больше волнует физика явления. Если Вы уверены в Вашей интерпретации, то, конечно же, я подправлю ответ. Я в физике не силен.
Мне, честно говоря, непонятен Ваш подход к решению задачи скорее в математическом, нежели в физическом отношении. Непонятны, в частности, выкладки из которых Вы нашли число k. Впрочем, смотрите сами: Вы администратор рассылки.
Что касается физики процесса, то, на мой взгляд, для осуществления процесса горения нужна совокупность трёх факторов, о чём я указал в своём ответе: горючего, окислительной среды и источника начала реакции - искры.
Предлагаю Вам в качестве арбитра привлечь Романа Чаплинского. Он по образованию химик, в математике тоже разбирается.
Скорее всего, нужны все 3 фактора. Более того, искра приведёт к возгоранию только если произойдёт после утечки топлива и попадания воздуха (или прибор будет искрить всё время, пока не загорится). По крайней мере искра зажечь топливо без кислорода неспособна точно. Может ли загореться воздушно-топливная смесь без искры? Только если там вдруг окажется какая-либо другая сильно нагретая поверхность (и то не гарантировано). Так что тоже нет.
А ещё я не уверен, что эта формула работает так, как надо: Р(А1+...+Аk)=k*Р(А)-Р(А1*...*Аk)=0,00063k-(0,00063)^k=0,00063(k-0,00063^(k-1)) Почему тут вычитается только вероятность возгорания сразу всех насосов, а не нескольких. И при большей вероятности и значительном количестве эта формула легко перевалит за 100%
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Гордиенко Андрей Владимирович