Tambayoyi Misali Game da Dokar Ƙara Abubuwa Biyu A da B waɗanda Ba Su Keɓance Juna Ba
Pendahuluan
A cikin yiwuwar, fahimtar muhimman ra'ayoyi da aikace-aikacensu yana da matuƙar muhimmanci. Wani ra'ayi da ke tasowa akai-akai shine ƙa'idar ƙara yiwuwar. A cikin wannan labarin, za mu zurfafa cikin ƙa'idar ƙara don abubuwan da ba na musamman ba guda biyu. Abubuwan da ba na musamman ba suna faruwa ne lokacin da abubuwan da suka faru guda biyu suka raba wani abu mai yiwuwa, ko kuma a wata ma'anar, dukkan abubuwan da suka faru za su iya faruwa a lokaci guda. Don sauƙaƙe fahimta, bari mu dubi matsalar misali da bayaninta.
Ka'idar Asali
Idan muna da abubuwa biyu na A da B waɗanda ba su da bambanci a tsakaninsu, to za a iya bayyana ƙa'idar ƙarawa don ƙididdige yiwuwar aukuwar A ko B kamar haka:
\[ P(A \kofin B) = P(A) + P(B) – P(A \kofin B) \]
Nan:
– \( P(A \cup B) \) shine yuwuwar cewa A ko B (ko duka biyun) sun faru.
– \( P(A) \) shine yuwuwar faruwar lamarin A.
– \( P(B) \) shine yuwuwar faruwar lamarin B.
– \( P(A \cap B) \) shine yuwuwar cewa A da B suna faruwa a lokaci guda.
Misalin matsalolin
Tambaya ta 1: A makaranta, kashi 70% na ɗalibai suna son yin ƙwallon ƙafa (taron A), kashi 40% na ɗalibai suna son yin ƙwallon kwando (taron B), kuma kashi 20% na ɗalibai suna son yin wasanni biyu. Menene yuwuwar ɗalibi ya so yin ƙwallon ƙafa ko ƙwallon kwando?
Tattaunawa:
An sani:
– \( P(A) = 0.70 \)
– \( P(B) = 0.40 \)
– \( P(A \cap B) = 0.20 \)
Amfani da ƙa'idar ƙara abubuwa biyu masu ban sha'awa:
\[ P(A \kofin B) = P(A) + P(B) – P(A \kofin B) \]
\[ P(A \kofin B) = 0.70 + 0.40 – 0.20 \]
\[ P(A \kofin B) = 0.90 \]
Don haka, yuwuwar ɗalibi ya so yin wasan ƙwallon ƙafa ko ƙwallon kwando shine 0.90 ko 90%.
Tambaya ta 2: A wani bincike, kashi 35% na masu amsa suna son kofi (taron A), kashi 50% na masu amsa suna son shayi (taron B), kuma kashi 10% na masu amsa suna son duka biyun. Menene yuwuwar cewa mai amsa yana son kofi ko shayi?
Tattaunawa:
An sani:
– \( P(A) = 0.35 \)
– \( P(B) = 0.50 \)
– \( P(A \cap B) = 0.10 \)
Amfani da ƙa'idar ƙara abubuwa biyu masu ban sha'awa:
\[ P(A \kofin B) = P(A) + P(B) – P(A \kofin B) \]
\[ P(A \kofin B) = 0.35 + 0.50 – 0.10 \]
\[ P(A \kofin B) = 0.75 \]
Don haka, yuwuwar cewa wanda ake amsawa yana son kofi ko shayi shine 0.75 ko 75%.
Tambaya ta 3: Daga bayanan kididdiga, an san cewa kashi 60% na mutane suna son karanta littattafai (taron A), kashi 30% na mutane suna son kallon fina-finai (taron B), kuma kashi 15% na mutane suna son yin duka biyun. Menene yuwuwar mutum ya so karanta littattafai ko kallon fina-finai?
Tattaunawa:
An sani:
– \( P(A) = 0.60 \)
– \( P(B) = 0.30 \)
– \( P(A \cap B) = 0.15 \)
Amfani da ƙa'idar ƙara abubuwa biyu masu ban sha'awa:
\[ P(A \kofin B) = P(A) + P(B) – P(A \kofin B) \]
\[ P(A \kofin B) = 0.60 + 0.30 – 0.15 \]
\[ P(A \kofin B) = 0.75 \]
Don haka, yuwuwar mutum ya son karanta littattafai ko kallon fina-finai shine 0.75 ko 75%.
Tambaya ta 4: Daga wani bincike da aka gudanar a wani gidan cin abinci, kashi 55% na abokan ciniki sun yi odar pizza (taron A), kashi 35% na abokan ciniki sun yi odar taliya (taron B), kuma kashi 20% na abokan ciniki sun yi odar duka biyun. Menene yuwuwar abokin ciniki ya yi odar pizza ko taliya?
Tattaunawa:
An sani:
– \( P(A) = 0.55 \)
– \( P(B) = 0.35 \)
– \( P(A \cap B) = 0.20 \)
Amfani da ƙa'idar ƙara abubuwa biyu masu ban sha'awa:
\[ P(A \kofin B) = P(A) + P(B) – P(A \kofin B) \]
\[ P(A \kofin B) = 0.55 + 0.35 – 0.20 \]
\[ P(A \kofin B) = 0.70 \]
Don haka, yuwuwar abokin ciniki ya yi odar pizza ko taliya shine 0.70 ko 70%.
Kammalawa
A cikin nazarin yiwuwar, ƙa'idar ƙara abubuwa biyu marasa keɓancewa A da B yana da matuƙar muhimmanci don tantance yuwuwar faruwar ɗaya ko duka biyun a lokaci guda. Amfani da dabarar \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \) yana sauƙaƙa ƙididdige yiwuwar haɗuwar abubuwan biyu waɗanda ke da mahadar hanya ko wani abu na gama gari.
Ta hanyar misalan da ke sama, za mu iya ganin cewa fahimtar matakan farko na wannan lissafi yana ba mu damar amfani da shi a yanayi daban-daban, ko a cikin ilimi, bincike, ko rayuwar yau da kullun. Bayyanannu da daidaito wajen gane waɗanne abubuwan da suka faru suna da mahimmanci don ƙididdige yiwuwar daidai. Saboda haka, yin aiki akai-akai da fahimtar mahimman ra'ayoyi za su taimaka sosai wajen fahimtar wannan batu.
