From 4e332204bb32ce14e68e4811558ad14d390d5c0e Mon Sep 17 00:00:00 2001 From: nik Date: Mon, 29 Sep 2025 08:16:27 +0300 Subject: [PATCH] initial commit --- lects/lec1/theory.pdf | Bin 0 -> 1765806 bytes lects/lec2/theory.pdf | Bin 0 -> 351374 bytes lects/lec3/theory.pdf | Bin 0 -> 278135 bytes pracs/prac1/tasks.pdf | Bin 0 -> 737833 bytes pracs/prac3/tasks.pdf | Bin 0 -> 715520 bytes pracs/prac4/ans.pdf | Bin 0 -> 51426 bytes pracs/prac4/ans.typ | 178 ++++++++++++++++++++++++++++++++++++++++++ pracs/prac4/tasks.pdf | Bin 0 -> 716097 bytes 8 files changed, 178 insertions(+) create mode 100644 lects/lec1/theory.pdf create mode 100644 lects/lec2/theory.pdf create mode 100644 lects/lec3/theory.pdf create mode 100644 pracs/prac1/tasks.pdf create mode 100644 pracs/prac3/tasks.pdf create mode 100644 pracs/prac4/ans.pdf create mode 100644 pracs/prac4/ans.typ create mode 100644 pracs/prac4/tasks.pdf diff --git a/lects/lec1/theory.pdf b/lects/lec1/theory.pdf new file mode 100644 index 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zdPpS#gme(_Vq)Q>3^YKRS#AI+dB3qH8lbItHDA9vD$9V{#PhqMpZ0QC|k# z1TCenP?teRJ1r`dF-;h#(}q6@ehj852U$_Bj7c}6cdE4pMOfsSLUs0X4mSHy+QV3>ERYmS^!ksrgPlRem|s-kXerkQCc`B>(VAmrs^ zb2y#K5X_zQHRW`c&&-6Q=r3$@v(Eg?%{$qgZO(y5C*)*7gF{&xiAB5!0bitor6ihH zL=qn|0!l)jLLqv55lM}w5SQpsAq5&B4!IORgYNo6LzHW}=_(De8!D(%6p69?V}hdJ MTt!dMakA6@0hT@${Qv*} literal 0 HcmV?d00001 diff --git a/pracs/prac4/ans.typ b/pracs/prac4/ans.typ new file mode 100644 index 0000000..11ccbbd --- /dev/null +++ b/pracs/prac4/ans.typ @@ -0,0 +1,178 @@ +#set text(size: 1.3em) +#set page(numbering: "1") +#set box( + fill: luma(195), + inset: (x: 4pt, y: 0pt), + outset: (y: 3pt), + radius: 2pt, +) + +#align(center)[= _Домашняя работа №4_] + +=== Q. _15. Два стрелка делают по одному выстрелу в мишень. Вероятность попадания первого стрелка равна 0,7, а второго — 0,8. Найти вероятность того, что мишень будет поражена. А если стрелки сделают по два выстрела?_ + +*A*: 1. Если стрелки делают один выстрел: + +Если мишень поражена, то значит хотя бы один попал. Найдем вероятность того, что оба не попали и вычтем из единицы: + +$P("оба не попали за один выстрел") = (1 - 0,7) dot (1 - 0,8) = 0,3 dot 0,2 = 0,06$, + +$P("мишень поражена за один выстрел") = 1 - P("оба не попали за один выстрел") = 1 - 0,06 = 0,94$ + +2. Если стрелки делают по два выстрела каждый: + +Будем следовать такой же логике и найдем вероятность что никто не попадет за два выстрела. + +$P("ни одного попадания") = P("оба не попали за один выстрел")^2 = 0,06^2 = 0,0036$, + +$P("мишень поражения за два выстрела") = 1 - P("ни одного попадания") = 1 - 0,0036 = 0,9964$ + +*Ответ*: #box()[$0,94; 0,9964$] + +#line(length: 100%) + +=== Q. _16. Станция метрополитена оборудована тремя эскалаторами. Вероятность безотказной работы для первого эскалатора равна 0,9; для второго – 0,8; для третьего – 0,7. Найти вероятность того, что произойдет поломка не более одного эскалатора._ + +*A*: Условие можно трактовать следующим образом: произойдет поломка либо одного, либо нуля эскалаторов. Рассмотрим каждый случай и сложим вероятности: + +Если сломался только один эскалатор, значит остальне два не сломались, поэтому рассмотрим все три случая: + +$ +P("поломка одного") = \ += underbrace((1 - 0\,9) dot 0\,8 dot 0\,7, "сломался первый") + underbrace(0\,9 dot (1 - 0\,8) dot 0\,7, "сломался второй") + underbrace(0\,9 dot 0\,8 dot (1 - 0\,7), "сломался третий") = \ += 0,1 dot 0,8 dot 0,7 + 0,9 dot 0,2 dot 0,7 + 0,9 dot 0,8 dot 0,3 = 0,398 +$ + +Для нуля сломанных эскалаторов получаем: + +$ +P("ничего не сломалось") = 0,9 dot 0,8 dot 0,7 = 0,504 +$ + +В результате получим: + +$ +P("поломка одного") + P("ничего не сломалось") = 0,398 + 0,504 = 0,902 +$ + +*Ответ*: #box()[$0,902$] + +#line(length: 100%) + +=== Q. _17. Один студент выучил 20 из 25 вопросов программы, а второй — только 15. Каждому из них задают по одному вопросу. Найти вероятность того, что правильно ответят: a. оба студента; b. только первый; c. только один из них; d. хотя бы один из студентов._ + +*A*: + +a) Чтобы верно ответили оба студента, нужно чтобы каждому попался вопрос из списка выученных, соответственно: + +$ +P("a)") = (20)/(25) dot (15)/(25) = (12)/(25) = 0,48 +$ + +b) Если правильно отвечает только первый, значит второй ошибается. Рассчитаем соответствующую вероятность: + +$ +P("b)") = (20)/(25) dot (25 - 15)/(25) = 8/(25) = 0,32 +$ + +c) Сложим два варианта: только первый + только второй: + +$ +P("c)") = P("b)") + (25 - 20)/(25) dot (15)/(25) = 0,32 + 3/(25) = 0,32 + 0,12 = 0,44 +$ + +d) Вычтем из единицы вероятность того, что никто не знал: + +$ +P("d)") = 1 - (25 - 20)/(25) dot (25 - 15)/(25) = 1 - 2/(25) = 0,08 +$ + + +*Ответ*: #box()[a) $0,48$; b) $0,32$; c) $0,44$; d) $0,08$] + +#line(length: 100%) + +=== Q. _18. Студент знает 30 из 40 вопросов программы. Экзаменатор задает вопросы до тех пор, пока не обнаружит пробел в знаниях студента. Найти вероятность того, что будут заданы: a. два вопроса; b. более двух вопросов; c. менее пяти вопросов._ + +*A*: a) Студент ответит на первый вопрос и ошибется на втором: + +$ +P("a)") = (30)/(40) dot (10)/(39) = 5/(26) +$ + +b) Значит студента точно ответит правильно на первые два вопроса: + +$ +P("b)") = (30)/(40) dot (29)/(39) = (29)/(52) +$ + +c) Вычтем из единицы вероятность того что понадобится не менее пяти вопросов: + +$ +P("c)") = 1 - (30)/(40) dot (29)/(39) dot (28)/(38) dot (27)/(37) = (12797)/(18278) +$ + +*Ответ*: #box(inset: (x: 4pt, y: 3pt))[a) $5/(26)$, b) $(29)/(52)$, c) $(12797)/(18278)$] + +#line(length: 100%) + +=== Q. _19. Вероятность дозвониться с первой попытки в справочное бюро вокзала равна 0,4. Какова вероятность того, что: a. удастся дозвониться при втором звонке; b. придется звонить не более трех раз?_ + +*A*: a) Значит мы не дозвонимся на первом и дозвонимся на втором: + +$ +P("a)") = (1 - 0,4) dot 0,4 = 0,24 +$ + +b) Мы либо дозвонимся на первой, либо на второй, либо на третьей попытке: + +$ +P("на первой") = 0,4 \ +P("на второй") = (1 - 0,4) dot 0,4 = 0,24 \ +P("на третьей") = (1 - 0,4)^2 dot 0,4 = 0,144 \ +P("b)") = 0,4 + 0,24 + 0,144 = 0,784 +$ + +*Ответ*: #box()[a) $0,24$, b) $0,784$] + +#line(length: 100%) + +=== Q. _20. Из колоды в 36 карт наудачу вынимают три карты. Какова вероятность того, что среди них будет хотя бы одна шестерка?_ + +*A*: Найдем вероятность того, что не выпадет ни одна шестерка и вычтем из единицы: + +$ +P("ни одной шестерки") = (32)/(36) dot (31)/(35) dot (30)/(34) = (248)/(357) +$ + +$ +P("хотя бы одна шестерка") = 1 - P("ни одной шестерки") = 1 - (248)/(357) = (109)/(357) +$ + +*Ответ*: #box(inset: (x: 4pt, y: 4pt))[$(109)/(357)$] + +#line(length: 100%) + +=== Q. _21. На АТС могут поступить вызовы трех типов. Вероятности поступления вызовов 1-го, 2-го и 3-го типа соответственно равны 0,2, 0,3, 0,5. поступило три вызова. Какова вероятность того, что a. все они разных типов; b. среди них нет вызова 2-го типа?_ + +*A*: + +*Ответ*: a) Разберем все варианты: + +#align(center)[#table(columns: 3)[I звонок][II звонок][III звонок][1][2][3][1][3][2][2][3][1][2][1][3][3][2][1][3][1][2]] + +Получаем: + +$ +P("a)") = (0,2 dot 0,3 dot 0,5) dot (3!) = 9/(50) = 0,18 +$ + +b) Значит каждый вызов это либо 1, либо 3: + +$ +P("b)") = (0,2 + 0,5)^3 = (343)/(1000) = 0,343 +$ + +*Ответ*: #box()[a) $0,18$; b) $0,343$] + + diff --git a/pracs/prac4/tasks.pdf b/pracs/prac4/tasks.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2710cfb86035a0b1a08a5a8cbbc758abda8dd2ba GIT binary patch literal 716097 zcmYg#V_;-Uv~?zSCblNFZQHgrv2EL&*qPY2I<{@w6MdQc?tSn3(|xM?RPEY(ueEkn zHHo~iC=CNG8#FXC0X>1O;ZJCC31br*CkrQc3TP7>xw)a~ zL@lhGOdJU4M6C>*OoUC0Y>iEz>7-3;%$&>#*q9g?+5c-X5HK;bF>TyAW!Ft1XnSbEcUlyK1b$okO($lE$8S)q_SA z6S!O!yT=0%YkVN)9r!)W25$p)#GVJIU_hC0@9nEt!?JD59tUa|e4Kf^QN@do;RrPJ 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