Probability — classical, empirical, axiomatic (Kolmogorov axioms)

2.7.5 · Maths › Statistics & Probability — Intermediate

HUM KYA MEASURE kar rahe hain?

Sets kyun? Kyunki "aur", "ya", "nahi" ban jaate hain intersection $\cap$ , union $\cup$ , complement $A^{c}$ . Logic ↔ set algebra, isliye hum events ke saath compute kar sakte hain.

1. Classical probability

Yeh kyun kaam karta hai: agar har outcome equally likely hai, toh har ek ki probability $\frac{1}{n}$ hai (unhe 1 tak sum hona chahiye). Event $A$ apne $m$ outcomes ka union hai, isliye $P (A) = m \cdot \frac{1}{n}$ .

Kaise / limitation: "equally likely" justify karne ke liye ek symmetry argument chahiye. Bent coin ke liye, ya "kya baarish hogi?" ke liye fail karta hai (count karne ke liye koi symmetric outcomes nahi hain).

2. Empirical (frequentist) probability

Limit kyun: chhote runs noisy hote hain (10 tosses mein 7 heads aa sakte hain). Law of Large Numbers kehta hai ki relative frequency ek fixed number ki taraf settle down karti hai jaise jaise $N$ badhta hai — woh stable value hi probability hai.

3. Axiomatic probability (Kolmogorov, 1933)

Kyun zaroorat hai: classical symmetry assume karta hai; empirical ko infinite trials chahiye. Kolmogorov ne iske bajaye poocha: koi bhi "probability" ke liye kaunse minimal rules zaroor satisfy hone chahiye? Tab classical aur empirical dono is framework ke andar numbers assign karne ke valid tarike ban jaate hain.

Baaki sab kuch in teeno se derive hota hai — koi extra assumption nahi.

Scratch se Derivations

Figure — Probability — classical, empirical, axiomatic (Kolmogorov axioms)

Worked examples jo sab kuch ek saath jodte hain

Common mistakes (steel-manned)

Flashcards

Classical probability formula aur uski key assumption

P (A) = m / n

; require karta hai ki saare

n

outcomes equally likely hon.

Empirical probability ki definition

P (A) = lim_{N \to \infty} f / N

, yaani bahut saare trials mein relative frequency.

Teen Kolmogorov axioms batao

(1)

P (A) \geq 0

; (2)

P (S) = 1

; (3) disjoint events add hote hain:

P (\cup A_{i}) = \sum P (A_{i})

P (\emptyset) = 0

prove karo

P (S) = P (S \cup \emptyset) = P (S) + P (\emptyset) \Rightarrow P (\emptyset) = 0

Complement rule aur uska proof

P (A^{c}) = 1 - P (A)

;

A, A^{c}

disjoint hain,

A \cup A^{c} = S

, isliye

P (A) + P (A^{c}) = 1

General addition rule

P (A \cup B) = P (A) + P (B) - P (A \cap B)

Addition rule mein

P (A \cap B)

kyun subtract karte hain

Overlap dono

P (A)

aur

P (B)

mein count hota hai, isliye ek baar hatate hain.

Axioms se

P (A) \leq 1

kyun

P (A^{c}) = 1 - P (A) \geq 0 \Rightarrow P (A) \leq 1

Empirical probability ko kaunsa theorem justify karta hai

Law of Large Numbers — relative frequency true probability ki taraf converge karti hai.

Gambler's fallacy — fix

Independent trials ki koi memory nahi hoti; LLN early data ko swamp karta hai, compensate nahi karta.

Recall Feynman: ek 12-saal ke bachche ko samjhao

Probability ek "kitna likely score" hai 0 (kabhi nahi) se 1 (hamesha) tak. Score pane ke teen tarike: (1) Gino — die ke 6 equal sides hain, toh har side ka score $1/6$ hai. (2) Bahut baar try karo — ek weird bottle cap 100 baar uchhalo, landing gino, woh fraction score hai. (3) Rules — ek samajhdaar mathematician (Kolmogorov) ne kaha: scores kabhi negative nahi hote, "kuch hoga" ka score 1 hai, aur jo cheezein saath nahi ho sakti unke scores seedha add ho jaate hain. In 3 chhote rules se tum har probability fact nikaal sakte ho, jaise jaadu.

Connections

Conditional Probability & Bayes' Theorem — $P (A ∣ B)$ seedha inhi axioms pe build karta hai.
Independence of Events — $P (A \cap B) = P (A) P (B)$ se define hota hai.
Set Theory — Union, Intersection, Complement — woh algebra jisme events rehte hain.
Law of Large Numbers — empirical se axiomatic tak ka bridge.
Random Variables & Probability Distributions — sample space pe functions.
Permutations & Combinations — classical probability ke peeche counting engine.