Zitsanzo za Mafunso Okhudza Kuthekera kwa Zochitika Zophatikizana
Chiyambi cha Kuthekera kwa Zochitika Zophatikizana
Kuthekera ndi nthambi ya masamu yomwe imaphunzira kuthekera kwa chochitika kuchitika. Kuthekera kwa chochitika chophatikizana ndi kuthekera kwa chochitika chomwe chili ndi zochitika zoposa chimodzi. Mwachitsanzo, kuthekera kogubuduza nambala yofanana pa die ndi ace kuchokera pa deck ya makadi osewerera ndi zitsanzo za zochitika zophatikizana. Nkhaniyi ikambirana mavuto angapo a zitsanzo ndikukambirana za kuthekera kwa zochitika zophatikizana.
Lingaliro Loyambira la Kuthekera kwa Zochitika Zophatikizana
Pali mitundu iwiri ya zochitika zosakanikirana:
1. Zochitika Zapadera: Zochitika ziwiri zomwe sizingachitike nthawi imodzi. Mwachitsanzo, pozungulira die, zochitika za kuzungulira 2 ndi 5 ndi zochitika zomwe sizingachitike nthawi imodzi chifukwa sizingatheke kuzungulira manambala onse awiri nthawi imodzi.
2. Zochitika Zosagwirizana: Zochitika ziwiri zomwe zingachitike nthawi imodzi. Mwachitsanzo, pojambula makadi osewerera, zochitika zopeza khadi la mtima (♥) ndi khadi lokhala ndi nambala 10 ndi zochitika zomwe sizigwirizana chifukwa pali khadi la mtima lokhala ndi nambala 10.
Nazi njira zina zoyambira zomwe zimagwiritsidwa ntchito powerengera mwayi wa zochitika zophatikizika:
– P(A kapena B) (pa zochitika zomwe sizimachitika nthawi imodzi): \(P(A \cup B) = P(A) + P(B) – P(A \cap B)\)
– P(A kapena B) (pazochitika zomwe sizigwirizana): \(P(A \cup B) = P(A) + P(B)\)
– P(A ndi B) (pazochitika zodziyimira pawokha): \(P(A \cap B) = P(A) \times P(B)\)
Mafunso ndi Kukambirana Zitsanzo
Chitsanzo Funso 1: Dayisi
Funso:
Kodi mwayi wopeza nambala yofanana kapena nambala yoposa 4 pa die ndi wotani?
Kukambirana:
Choyamba, tiyeni tifotokoze zochitika:
– Chochitika A: Kupeza nambala yofanana (2, 4, 6)
– Chochitika B: Kupeza nambala yoposa 4 (5, 6)
Kenako, tikudziwa mwayi wa chochitika chilichonse:
– \(P(A) = \frac{3}{6} = \frac{1}{2}\)
– \(P(B) = \frac{2}{6} = \frac{1}{3}\)
Popeza pali nambala 6 yomwe ikuphatikizidwa mu zochitika zonse ziwiri A ndi B, tifunika kuwerengera \(P(A \cap B)\):
– \(P(A \cap B) = \frac{1}{6}\) (chifukwa nambala imodzi yokha, yomwe ndi 6, ikuphatikizidwa mu A ndi B)
Pogwiritsa ntchito njira ya zochitika zomwe sizimachitika nthawi imodzi:
\[P(A \cup B) = P(A) + P(B) – P(A \cap B) = \frac{1}{2} + \frac{1}{3} – \frac{1}{6}\]
Tiyeni tipange ma denominator a magawo awa kukhala ofanana:
\[P(A \cup B) = \frac{3}{6} + \frac{2}{6} – \frac{1}{6} = \frac{4}{6} = \frac{2}{3}\]
Kotero, mwayi wopeza nambala yofanana kapena nambala yoposa 4 ndi \(\frac{2}{3}\).
Chitsanzo cha Funso 2: Makhadi Osewerera
Funso:
Kodi mwayi wopeza Ace kapena spade kuchokera pa bolodi la makadi osewerera ndi wotani?
Kukambirana:
Choyamba, tiyeni tifotokoze zochitika:
- Chochitika A: Kupeza khadi la Ace (4 onse, limodzi pa suti iliyonse)
- Chochitika B: Kupeza khadi la spade (onse 13)
Kenako, tikudziwa mwayi wa chochitika chilichonse:
– \(P(A) = \frac{4}{52} = \frac{1}{13}\)
– \(P(B) = \frac{13}{52} = \frac{1}{4}\)
Popeza Ace of Spades ikuphatikizidwa mu zochitika zonse ziwiri A ndi B, tifunika kuwerengera \(P(A \cap B)\):
– \(P(A \cap B) = \frac{1}{52}\)
Pogwiritsa ntchito njira ya zochitika zomwe sizimachitika nthawi imodzi:
\[P(A \cup B) = P(A) + P(B) – P(A \cap B) = \frac{1}{13} + \frac{1}{4} – \frac{1}{52}\]
Tiyeni tipange ma denominator a magawo awa kukhala ofanana:
\[
P(A \cup B) = \frac{4}{52} + \frac{13}{52} – \frac{1}{52} = \frac{16}{52} = \frac{4}{13}
\]
Kotero, mwayi wokhala ndi Ace kapena spade ndi \(\frac{4}{13}\).
Chitsanzo Chachitatu: Mpira mu Bokosi
Funso:
Mu bokosi muli mipira itatu yofiira, mipira inayi yabuluu, ndi mipira isanu yobiriwira. Ngati mpira umodzi watengedwa mwachisawawa, kodi mwayi wopeza mpira wofiira kapena mpira wobiriwira ndi wotani?
Kukambirana:
Choyamba, tiyeni tifotokoze zochitika:
– Chochitika A: Kupeza mpira wofiira (nambala 3)
– Chochitika B: Kupeza mpira wobiriwira (nambala 5)
Kenako, tikudziwa mwayi wa chochitika chilichonse:
– Chiwerengero chonse cha mipira = 3 + 4 + 5 = 12
– \(P(A) = \frac{3}{12} = \frac{1}{4}\)
– \(P(B) = \frac{5}{12}\)
Popeza palibe mpira womwe ungakhale wofiira ndi wobiriwira nthawi imodzi, zochitika izi ndizosiyana:
\[P(A \cup B) = P(A) + P(B) = \frac{1}{4} + \frac{5}{12}\]
Tiyeni tipange ma denominator a magawo awa kukhala ofanana:
\[
P(A \cup B) = \frac{3}{12} + \frac{5}{12} = \frac{8}{12} = \frac{2}{3}
\]
Kotero, mwayi wopeza mpira wofiira kapena mpira wobiriwira ndi \(\frac{2}{3}\).
Chitsanzo Funso 4: Ndalama Ziwiri
Funso:
Ngati ndalama ziwiri zaponyedwa nthawi imodzi, kodi pali mwayi wotani woti mutu umodzi uwonekere?
Kukambirana:
Timatanthauzira Chochitika A: kukumana ndi chithunzi chimodzi kapena zingapo.
Pali zotsatira zinayi zomwe zingatheke poponya ndalama ziwiri:
1. HH
2. HT
3. TH
4. TT
Zochitika zomwe zili ndi chithunzi chimodzi kapena zingapo ndi izi:
- HT
– TH
– TT
Tiyeni tiwerengere mwayi wa chilichonse:
– Chiwerengero cha zochitika zomwe zingatheke (zonse): 4
- Chiwerengero cha zochitika zomwe zili ndi chithunzi chimodzi: 3
\[
P(A) = \frac{Chiwerengero cha zochitika zokhala ndi mutu umodzi osachepera}{Chiwerengero chonse cha zochitika} = \frac{3}{4}
\]
Kotero, mwayi woti chithunzi chimodzi chiwonekere ndi \(\frac{3}{4}\).
Mapeto
Kukambirana za mavuto omwe ali pamwambapa kukuwonetsa momwe tingawerengere mwayi wa chochitika chophatikizana, kaya ndi chosiyana kapena chosagwirizana. Mwa kumvetsetsa mfundo zoyambira ndikugwiritsa ntchito njira zolondola, titha kudziwa mwayi wa kuphatikiza zochitika zina zomwe zimachitika m'mikhalidwe yosiyanasiyana ya tsiku ndi tsiku. Pitirizani kuchita masewera olimbitsa thupi ndi mavuto osiyanasiyana kuti mukhale aluso kwambiri pakuzindikira mwayi wa zochitika zophatikizana.
