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0 این فرمول را میتوان به صورت زیر نوشت:
(P(AB) = P(B) P(A│B
که آن قانون ضرب احتمالها گوییم. به همین نحو ، احتمال شرطی B به شرط A را میتوان به صورت زیر بیان کرد:
(P(B│A) = P(AB)/P(A
که منجر به رابطه (P(AB) = P(A) P(B|A میشود. بنابراین قانون ضرب احتمالها این تساوی را بیان میکند که حاصلضرب احتمال شرطی یک پیشامد در احتمال پیشامد شرطی کننده ، برابر است با احتمال اشتراک آن دو پیشامد.]]>
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m داشته باشیم و n کبوتر در m لانه قرار بگیرند، آنگاه حداقل یک لانه وجود دارد که در آن بیش از یک کبوتر قرار گرفته باشد. به عنوان مثال از بین هر ۱۱ عدد طبیعی (که مقدار آنها را نمیدانیم.) رقم یکان حداثل دو عدد از این اعداد، یکسان است.
ترکیبیات، ریاضیات انتخاب یا آنالیز ترکیبی یکی از شاخههای جذاب ریاضیات است که به بررسی مسائل شمارش، گرافها، بازیها و نیز مسائل ساختاری روی مجموعهها متناهی میپردازد. از جمله کاربردهای مهم این شاخه میتوان به استفاده آن در برنامهنویسی کامپیوتر و الگوریتمها اشاره کرد. یکی از مسائلی که ترکیبیات را از دیگر شاخههای ریاضی متمایز میکند این است که آموختن آن نیاز به اطلاعات خاصی از ریاضیات ندارد و داشتن معلومات ریاضی دوره راهنمایی نیز برای درک آن کافی به نظر میرسد چرا که ریشههای ترکیبیات در واقع به مسائل معماگونه ریاضی و بازیها میرسد. بسیاری از مسائل ترکیبیات که در گذشته برای تفریح بررسی شدهاند امروزه اهمیت زیادی در ریاضیات محض و کاربردی دارند. در قرن اخیر ترکیبیات به یکی از مهمترین شاخههای ریاضیات تبدیل شده و مرزهای آن همواره گسترش پیدا میکند که یکی از مهمترین علل این گسترش سریع، اختراع کامپیوتر است: به علت سرعت بالای کامپیوترها بسیاری از مسائلی که قبلاً قابل بررسی نبودند، بررسی شدند. البته تقابل کامپیوتر و ترکیبیات یک طرفه نبودهاست و کامپیوترها نمیتوانستند مستقل عمل کنند و برای عمل نباز به برنامه داشتند. اساس برنامههای کامپیوتری غالباً الگوریتمهای ترکیبیاتی اند و به همین دلیل اهمیت و کاربرد ترکیبیات پس از اختراع کامپیوتر چندین برابر معلوم شد و باعث شد تا ریاضیدانان بسیاری به تحقیقات گسترده در این زمینه رو آوردند. مباحث ترکیبیات بسیار گستردهاند ولی اساس آن بر پایه روشهای شمارش است که از جمله این روشها میتوان به اصل جمع، اصل ضرب، جایگشت اشاره کرد. یکایک شمردن یا شمارش، ممکن است به عنوان فرآیندی آشکار تلقی شود که هر دانشجو در آغاز مطالعه علم حساب فرا میگیرد؛ ولی به نظر میرسد که پس از آن، به تدریج که دانشجو به زمینههای «دشوارتر» ریاضیات، چون جبر، هندسه، مثلثات، و حساب دیفرانسیل و انتگرال میرسد توجه بسیار کمتری به گسترش بیشتر مفهوم شمارش مبذول میشود. یکایک شمردن محدود به حساب نیست. کاربردهایی نیز در زمینههایی چون نظریه کدگذاری، حساب احتمالات، و آمار (در ریاضیات) و در تحلیل الگوریتمها (در علم کامپیوتر) دارد.]]>
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0 این فرمول را میتوان به صورت زیر نوشت:
(P(AB) = P(B) P(A│B
که آن قانون ضرب احتمالها گوییم. به همین نحو ، احتمال شرطی B به شرط A را میتوان به صورت زیر بیان کرد:
(P(B│A) = P(AB)/P(A
که منجر به رابطه (P(AB) = P(A) P(B|A میشود. بنابراین قانون ضرب احتمالها این تساوی را بیان میکند که حاصلضرب احتمال شرطی یک پیشامد در احتمال پیشامد شرطی کننده ، برابر است با احتمال اشتراک آن دو پیشامد.]]>
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