Misali na tambayar tattaunawa game da yuwuwar abubuwan da suka faru na mahaɗan da ba su da alaƙa da yanayin
Pendahuluan
Yiwuwa wani reshe ne na lissafi wanda ke nazarin yuwuwar faruwar wani abu. Ɗaya daga cikin muhimman ra'ayoyi a cikin ka'idar yuwuwar shine abubuwan da suka faru masu rikitarwa, waɗanda za a iya rarraba su a matsayin ko dai masu zaman kansu ko kuma waɗanda ke da sharaɗi. Wannan labarin zai bincika waɗannan ra'ayoyi sosai ta hanyar misalai da tattaunawa.
Abubuwan da suka Faru a Ciki
Wani abu mai hadewa hade ne na abubuwa biyu ko fiye da ke faruwa a cikin samfurin sarari. Akwai nau'ikan abubuwan hadewa guda biyu: abubuwan da suka kebanta da juna da kuma abubuwan da suka kebanta da yanayi.
1. Abubuwan da suka faru masu zaman kansu: Ana cewa abubuwa biyu suna da zaman kansu idan nasarar ko gazawar wani abu bai shafi sakamakon ɗayan ba. Misali, sakamakon jefa tsabar kuɗi da kuma jujjuyawar kuɗi.
2. Abubuwan da suka Sharaɗi: Abubuwan da suka sharaɗi suna faruwa ne lokacin da nasarar ko gazawar wani abu ya shafi yuwuwar faruwar wani abu. Misali, damar wani ya kamu da cuta idan yana da alaƙa da kwayar halittar wannan cuta.
Yiwuwar Abubuwan da Za Su Faru Masu Zaman Kansu
Tsarin lissafin yiwuwar aukuwar abubuwa biyu masu zaman kansu na A da B shine:
\[ P(A \cap B) = P(A) \times P(B) \]
Ina:
– \(P(A \cap B)\) shine yuwuwar abubuwan da suka faru A da B suna faruwa a lokaci guda.
– \(P(A)\) shine yuwuwar faruwar lamarin A.
– \(P(B)\) shine yuwuwar faruwar lamarin B.
Tambayoyi da Tattaunawa Misali: Abubuwan da Suka Faru Tsakanin Juna Masu Zaman Kansu
Tambaya ta 1: Ana jefa tsabar kuɗi sannan a mirgina wani abin da ke gefe shida. A ƙayyade yiwuwar samun kan tsabar kuɗi da kuma 4 a kan abin da ke ciki.
Tattaunawa:
– Yiwuwar hoton da ke bayyana a kan tsabar kuɗi: \( P(G) = \frac{1}{2} \)
– Yiwuwar samun lamba 4 a kan dabe: \( P(4) = \frac{1}{6} \)
Tunda tsabar kudin da kuma mutu abubuwa ne guda biyu masu zaman kansu, yiwuwar duka biyun su faru a lokaci guda shine:
\[ P(G \cap 4) = P(G) \times P(4) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12} \]
Don haka, yuwuwar samun hoto a kan tsabar kudin da kuma 4 a kan dice shine \( \frac{1}{12} \).
Yiwuwar Abubuwan da Suka Faru a Sharaɗi
Yiwuwar sharaɗin abubuwan da suka faru guda biyu A da B shine yuwuwar cewa A ta faru tunda B ta faru. An nuna dabarar ƙididdige yiwuwar sharaɗi kamar haka:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Ina:
– \( P(A|B) \) shine yuwuwar faruwar lamarin A muddin lamarin B ya riga ya faru.
– \( P(A \cap B) \) shine yuwuwar abubuwan da suka faru A da B suna faruwa a lokaci guda.
– \( P(B) \) shine yuwuwar faruwar lamarin B.
Tambayoyi da Tattaunawa Misali: Abubuwan da Suka Faru a Sharaɗi
Tambaya ta 2: Daga cikin akwati mai ɗauke da ƙwallo ja 3 da ƙwallo shuɗi 2, ana zana ƙwallo biyu ba zato ba tsammani ɗaya bayan ɗaya ba tare da maye gurbinsu ba. Nemo yiwuwar cewa ƙwallon da aka zana ta biyu ja ce ganin cewa ƙwallon farko ma ja ce.
Tattaunawa:
Misali:
– A shine abin da ya faru da cewa ƙwallon farko ta yi ja.
– B shine abin da ya faru da cewa ƙwallon ta biyu ta yi ja.
Muna neman \( P(B|A) \). Da farko, muna lissafin \(P(A)\) da \(P(A \cap B)\):
Jimlar adadin ƙwallo = 5 (ja 3 da shuɗi 2).
Damar farko ta ƙwallon ja:
\[
P(A) = \frac{3}{5}
\]
Bayan an zana ƙwallon ja ta farko, adadin sauran ƙwallon ja ya zama 2 kuma jimillar adadin ƙwallon ya zama 4.
Yiwuwar ƙwallon ta biyu ta zama ja bayan an ja ƙwallon farko:
\[
P(B|A) = \frac{2}{4} = \frac{1}{2}
\]
Don haka, yuwuwar cewa ƙwallon ta biyu ta yi ja idan ƙwallon farko ma ta yi ja shine \( \frac{1}{2} \).
Tambayoyin Misali na Haɗaka
Domin zurfafa fahimtarmu, za mu iya haɗa abubuwan da suka faru masu zaman kansu da waɗanda suka shafi yanayi a cikin tambaya ɗaya.
Tambaya ta 3: Jaka tana ɗauke da ƙwallo ja 5 da ƙwallo shuɗi 3. Ana zana ƙwallo biyu ba tare da an maye gurbinsu ba. Nemo yiwuwar cewa ƙwallo ta farko ja ce, ƙwallo ta biyu kuma shuɗi ce.
Tattaunawa:
Muna amfani da irin wannan alamar kamar yadda aka yi a baya:
– A shine abin da ya faru da cewa ƙwallon farko ta yi ja.
– B shine taron da ƙwallon ta biyu ta yi shuɗi.
Da farko, muna ƙididdige yiwuwar kowace aukuwa a jere.
Yiwuwar ƙwallon farko ta zama ja:
\[
P(A) = \frac{5}{8}
\]
Idan ƙwallon farko ja ce, adadin sauran ƙwallan ja = 4, da kuma jimillar adadin sauran ƙwallan = 7.
Yiwuwar ƙwallon ta biyu ta zama shuɗi bayan ƙwallon farko ta zama ja:
\[
P(B|A) = \frac{3}{7}
\]
Don haka, yuwuwar cewa ƙwallon farko ja ce, ƙwallon ta biyu kuma shuɗi ce sakamakon waɗannan yiwuwar sharaɗi guda biyu:
\[
P(A \cap B) = P(A) \times P(B|A) = \frac{5}{8} \times \frac{3}{7} = \frac{15}{56}
\]
Don haka, yuwuwar cewa ƙwallon farko ja ce, ƙwallon ta biyu kuma shuɗi ce.
Kammalawa
A cikin ka'idar yiwuwar, fahimtar bambanci tsakanin abubuwan da suka faru masu zaman kansu da na yanayi yana da mahimmanci don magance matsalolin da suka shafi abubuwan da suka faru masu rikitarwa. Ta hanyar misalan matsalolin, muna koyon yadda ake ƙididdige yiwuwar yanayi daban-daban da suka shafi waɗannan ra'ayoyi guda biyu. Kyakkyawan fahimtar waɗannan ra'ayoyi na iya taimakawa wajen yanke shawara a cikin yanayi na gaske kamar sarrafa haɗari, wasanni, da binciken kimiyya.
Amfani da lissafi a rayuwar yau da kullum yana nuna yadda wannan fahimta take da matuƙar muhimmanci ga fannoni daban-daban na rayuwar ɗan adam, tun daga mafi sauƙi har zuwa mafi rikitarwa. Ta hanyar ci gaba da yin aiki da kuma fahimtar waɗannan muhimman ra'ayoyi, ƙwarewarmu a nazarin yiwuwar za ta ƙaru.
