Ehtimollik

⭐ — imtihonda eng ko'p ishlatiladigan formulalar.

Klassik ta'rif

• ⭐ Hodisa $A$ ning ehtimoli (barcha natijalar teng imkoniyatli bo'lsa):

$P(A) = \frac{m}{n}$

bunda $n$ — barcha mumkin bo'lgan natijalar soni, $m$ — $A$ uchun qulay natijalar soni.

• ⭐ $0 \le P(A) \le 1$. Mumkin bo'lmagan hodisa uchun $P = 0$, muqarrar hodisa uchun $P = 1$. Javob $1$ dan katta yoki manfiy chiqsa — xato.
• ⚠️ Natijalar teng imkoniyatli bo'lsin. Ikkita tanga tashlanganda "gerb–raqam" va "raqam–gerb" alohida natijalar: jami $4$ ta ($GG, GR, RG, RR$), $3$ ta emas.

Qarama-qarshi hodisa

• ⭐ $P(\bar{A}) = 1 - P(A)$.
• ⭐ "Kamida bittasi" hodisasi uchun: $P = 1 - P(\text{birortasi ham})$. Masalan, tanga $4$ marta tashlanganda kamida bir marta raqam tushishi: $1 - \dfrac{1}{16} = \dfrac{15}{16}$.

Qo'shish va ko'paytirish qoidalari

• ⭐ Birgalikda bo'lmagan ("yoki") hodisalar: $P(A \text{ yoki } B) = P(A) + P(B)$.
• Birgalikda bo'lishi mumkin bo'lganda: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
• ⭐ Mustaqil ("va") hodisalar: $P(A \text{ va } B) = P(A) \cdot P(B)$.
  • Misol: nishonga tekkazish ehtimoli har otishda $\dfrac{3}{5}$. Ikkala otishda ham tegish: $\dfrac{3}{5} \cdot \dfrac{3}{5} = \dfrac{9}{25}$. Kamida bittasida tegish: $1 - \dfrac{2}{5} \cdot \dfrac{2}{5} = \dfrac{21}{25}$.
  • Aynan bittasida tegish: $2 \cdot \dfrac{3}{5} \cdot \dfrac{2}{5} = \dfrac{12}{25}$.
• Bog'liq hodisalar (masalan, shar qaytarilmaydi): $P(A \text{ va } B) = P(A) \cdot P(B \mid A)$, bunda $P(B \mid A)$ — $A$ ro'y bergan bo'lsa, $B$ ning ehtimoli. Qopda $3$ qizil va $2$ ko'k shar: ketma-ket ikkalasi qizil: $\dfrac{3}{5} \cdot \dfrac{2}{4} = \dfrac{3}{10}$.

Kombinatorika yordamida

• Guruh tanlash ("bir vaqtda $k$ ta olindi"): $P = \dfrac{C_m^{k}}{C_n^{k}}$ ko'rinishida. Masalan, $10$ ta sharda $4$ tasi qizil, $2$ ta olinsa ikkalasi qizil: $\dfrac{C_4^2}{C_{10}^2} = \dfrac{6}{45} = \dfrac{2}{15}$.
• Bu usul qaytarmasdan ketma-ket olishga teng: yuqoridagi misol $\dfrac{4}{10} \cdot \dfrac{3}{9} = \dfrac{2}{15}$.

Tez-tez uchraydigan holatlar

• ⭐ Tanga $n$ marta: barcha natijalar $2^n$. Aynan $k$ marta gerb: $\dfrac{C_n^k}{2^n}$. Masalan, $5$ marta, aynan $2$ gerb: $\dfrac{10}{32} = \dfrac{5}{16}$.
• ⭐ Ikki kubik (barcha natijalar $6 \cdot 6 = 36$). Yig'indi bo'yicha qulay natijalar:

Masalan, yig'indi $7$ bo'lish ehtimoli $\dfrac{6}{36} = \dfrac{1}{6}$. Bir kubik: juft ochko — $\dfrac{3}{6} = \dfrac{1}{2}$.

• Bernulli sxemasi: har urinishda muvaffaqiyat ehtimoli $p$ ($q = 1 - p$) bo'lsa, $n$ ta urinishda aynan $k$ ta muvaffaqiyat:

$P_n(k) = C_n^k \, p^k q^{\,n - k}$

Masalan, $p = \dfrac{1}{3}$, $n = 3$, $k = 2$: $3 \cdot \dfrac{1}{9} \cdot \dfrac{2}{3} = \dfrac{2}{9}$.

Geometrik ehtimollik

• Natijalar kesma, yuza yoki hajmdagi nuqta bo'lsa: $P = \dfrac{\text{qulay qismning o'lchami}}{\text{butun o'lcham}}$. Masalan, tomoni $2$ bo'lgan kvadratga tasodifiy nuqta tashlansa, unga ichki chizilgan birlik doiraga tushish ehtimoli $\dfrac{\pi \cdot 1^2}{2^2} = \dfrac{\pi}{4}$.

Tipik xatolar

• ⚠️ Ehtimollarni qo'shish kerak joyda ko'paytirish (va aksincha): "va" — ko'paytiring, "yoki" — qo'shing.
• ⚠️ Kubiklarni "bir xil" deb hisoblash: $(1, 2)$ va $(2, 1)$ ikki xil natija.
• ⚠️ "Kamida bitta"ni to'g'ridan-to'g'ri sanash; qarama-qarshi hodisadan foydalanish ancha oson.
• ⚠️ Qaytarib olish va qaytarmasdan olishni aralashtirish: ikkinchi holda shar soni kamayadi.