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1)^2, 4), frac(sqrt(3), 2) a^2) = frac(pi (sqrt(3) - 1)^2, 4) dot frac(2, sqrt(3)) = frac(pi (sqrt(3) - 1)^2, 2 sqrt(3)) = frac(pi(2 - sqrt(3)), sqrt(3)) approx 0.48 $ - *Ответ*: $0.48$ #align(center)[=== _Тема 3. Формула полной вероятности. Формула Байеса._] *3*. _Два стрелка стреляют по одному разу, независимо друг от друга выбирая одну из двух мишеней. Вероятность выбора первой мишени для них $0.5$ и $2/3$ соответственно, а вероятности попадания в первую мишень $0.8$ для первого стрелка и $0.9$ для второго стрелка, во вторую мишень соответственно $0.7$ и $0.8$. Какова вероятность хотя бы одного попадания в какую-либо мишень?_ +*Решение*: + +$ +P("хотя бы один попадет") = 1 - P("оба промахнутся") +$ + +Для первого стрелка: если он выбрал первую мишень ($1/2$), промах = $1 - 0.8 = 0.2$; если выбрал вторую ($1/2$), промах = $1 - 0.7 = 0.3$. + +Посчитав по формуле полной вероятности, получим: + +$ +P("промах первого") = 0.5 dot 0.2 + 0.5 dot 0.3 = 0.25 +$ + +Для второго стрелка: если он выбрал первую мишень ($2/3$), промах = $1 - 0.9 = 0.1$; если выбрал вторую ($1/3$), промах = $1 - 0.8 = 0.2$. + +Посчитав по формуле полной вероятности, получим: + +$ +P("промах второго") = 2/3 dot 0.1 + 1/3 dot 0.2 = frac(2, 15) +$ + +Вероятность того, что оба промахнутся: + +$ +P("оба промахнутся") = P("промах первого") dot P("промах второго") = 0.25 dot frac(2, 15) = frac(1, 30). +$ + +Тогда ответ: + +$ +P("хотя бы один попадет") = 1 - P("оба промахнутся") = 1 - frac(1, 30) = frac(29, 30) +$ + +*Ответ*: $frac(29, 30)$ + #line(length: 100%) *13*. _В трех одинаковых урнах находятся шары: в первой - с номерами от $1$ до $9$, во второй - от $10$ до $20$, в третьей - от $21$ до $30$ включительно. Из случайно выбранной урны берется шар, и оказывается, что его номер делится на $5$. Какова вероятность того, что этот шар взят из первой урны?_ +*Решение*: Пусть события $A_1, A_2, A_3$ - выбрана первая, вторая и третья урны соответственно. Так как урны выбираются случайно, то $P(A_1) = P(A_2) = P(A_3) = 1/3$. Пусть $B$ - событие "выпал номер, делящийся на 5". + +По формуле Байеса: + +$ +P(A_1 | B) = frac(P(B | A_1) P(A_1), P(B | A_1)P(A_1) + P(B | A_2) P(A_2) + P(B | A_3)P(A_3)) +$ + +Найдем $P(B | A_i)$. + +В урне 1 есть только одно число, кратное 5, то есть $P(B | A_1) = 1/9$ (так как всего чисел 9). + +В урне 2 есть 3 числа, кратных 5, то есть $P(B | A_2) = 3/11$ (всего чисел 11). + +В урне 3 есть 2 числа, кратных 5, то есть $P(B | A_3) = 2/(10) = 1/5$ (всего 10 чисел). + +При подстановке всего в формулу выше и опустив расчеты, получим: + +$ +P(A_1 | B) = frac(55, 289) +$ + +*Ответ*: $frac(55, 289)$ + #align(center)[=== _Тема 4. Схема Бернулли._] *3*. _Прибор содержит шесть однотипных микросхем, вероятность выхода из строя каждой в течение одного месяца равна $0.2$. Найти вероятность того, что в течение этого срока из строя выйдет не более половины микросхем._ +*Решение*: + +*Ответ*: + #line(length: 100%) *13*. Производится испытание на "самовозгорание" пяти телевизоров. Прогонка продолжается двое суток. За указанное время каждый из телевизоров перегревается и "самовозгорается" с вероятностью $0.1$. Найти вероятность того, что на момент окончания испытаний сгорит не более двух телевизоров. + +*Решение*: + +*Ответ*: