Probability — classical, empirical, axiomatic (Kolmogorov axioms): Visual walkthrough

2.7.5 · D2 · Maths › Statistics & Probability — Intermediate › Probability — classical, empirical, axiomatic (Kolmogorov ax

Yeh page parent note ka picture companion hai. Yeh Set Theory — Union, Intersection, Complement ki language pe rely karta hai.

Step 1 — Sample space ek equal tiles ka board hai

KYA HAI. Ek random experiment imagine karo. Har possible outcome ek chhota square tile hai. Unhe saare side by side rakh ke ek bada rectangle banao. Woh poora rectangle sample space hai, likha jaata hai $S$ — "jo bhi ho sakta hai, sab."

TILES KYUN. Hum probability ko area ke roop mein dekhna chahte hain. Agar har tile same size ka ho, toh tiles ginana aur area measure karna same baat hai — aur area dekhne mein aankhein achi hoti hain. Yeh classical idea ("equally likely outcomes") ko ek picture mein badal deta hai.

PICTURE. Neeche figure dekho: 36 tiles (do dice roll karne ke $6 \times 6$ outcomes — parent ka running example). Poora peach rectangle $S$ hai.

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

Yahan symbol $P (\cdot)$ ka matlab sirf "is region ka board ki total area mein kitna fraction hai" hai. Is page par $P$ ka yahi matlab hoga hamesha.

Step 2 — Ek event ek coloured region hai

KYA HAI. Koi yes/no sawaal chuno, jaise "kya dice-sum even hai?" Jahan jawab haan hai woh tiles ek region banate hain. Unhe colour karo. Woh coloured region ek event hai, aur hum use ek letter se name dete hain — maano $A$ .

KYUN. Ek sawaal ko ek shape mein badalna matlab ab hum logic ki jagah geometry kar sakte hain. " $A$ kitna likely hai?" ban jaata hai "board ka kitna fraction $A$ colour ka hai?"

PICTURE. Neeche magenta region $A$ .

Agar $A$ koi tile cover nahi karta, $P (A) = 0$ ; agar sab cover karta hai, $P (A) = 1$ . Yeh number kabhi $0$ se neeche nahi ja sakta (negative area nahi ho sakta) — yeh Kolmogorov ka non-negativity axiom hai, dikhta hua.

Step 3 — Do events: overlap hi poori kahani hai

KYA HAI. Ab usi board par ek doosra event $B$ (violet) colour karo. Kyunki $A$ aur $B$ dono ek hi board par regions hain, woh tiles share kar sakte hain.

KYUN. Lagbhag har real sawaal do events involve karta hai — "King ya Heart", "baarish ya aandhi". " $A$ ya $B$ " ke baare mein sochne ke liye pehle dekhna hoga ki unke regions ek doosre ke saath kaise hain. Exactly do possibilities hain: woh overlap karte hain ya nahi karte. Step 4 aasaan no-overlap case handle karta hai; Steps 5–6 overlap handle karte hain.

PICTURE. Magenta $A$ aur violet $B$ ek shared purple sliver ke saath jahan woh cross karte hain.

Step 4 — Aasaan case: koi overlap nahi, bas add karo

KYA HAI. Maano dono regions nahi milte (disjoint). Toh " $A$ ya $B$ " ka area find karne ke liye, bas dono areas add karo — koi tile double-book nahi hai.

KYUN. Jab kuch shared nahi hai, koi tile do baar count nahi hoti, toh areas cleanly add ho jaate hain. Yeh Kolmogorov ka additivity axiom ek picture ke roop mein dikhaya gaya hai: alag blobs, unke sizes add karo.

PICTURE. Do blobs jo nahi milte; unke areas ek "+" ke saath snap together hote hain.

Yeh poora rule sirf tab kaam karta hai jab woh nahi milte. Jis second woh milte hain, yeh line over-count karti hai — jo parent note ki pehli [!mistake] mein trap hai. Aage hum ise fix karte hain.

Step 5 — Overlap do baar count hota hai

KYA HAI. Regions ko phir se overlap karne do (Step 3 ki picture). Naive move try karo: $P (A) + P (B)$ add karo. Purple sliver dekho.

KYUN. $P (A)$ mein already purple sliver include hai (yeh $A$ ka part hai). $P (B)$ mein bhi wahi sliver include hai (yeh $B$ ka part hai). Toh jab aap $P (A) + P (B)$ likhte ho, sliver do baar measure hota hai — aapne ek baar $A$ ke andar aur ek baar $B$ ke andar us ke liye pay kiya.

PICTURE. Sliver double stripes se shaded hai, "counted 2×" flag ke saath.

Woh equation padhke, fix khud dikhai deta hai: sum exactly ek sliver zyada hai. Use subtract karo.

Step 6 — Overlap ek baar subtract karo: rule aa jaata hai

KYA HAI. Step-5 equation ko rearrange karo taaki jo chahiye woh isolate ho.

KYUN. Hum shared sliver ki ek extra copy hata dete hain taaki har tile exactly ek baar count ho. Woh single subtraction hi correction hai.

PICTURE. Sliver ki doosri copy "− $P (A \cap B)$ " arrow ke saath lift away ho jaati hai, clean union chhodke.

Step 4 ke saath consistency check. Agar events disjoint hain, $A \cap B = \emptyset$ , toh $P (A \cap B) = 0$ , aur box collapse ho jaata hai $P (A \cup B) = P (A) + P (B)$ mein — exactly aasaan case. Ek formula dono situations handle karta hai.

Step 7 — Degenerate & edge cases (taaki koi scenario surprise na kare)

KYA HAI & KYUN. Ek formula jis par aap trust karte ho use apne extreme inputs survive karne chahiye. Hum har corner ko tile picture par test karte hain.

Ek bhi corner boxed rule ko nahi todta — kyunki sliver ko ek baar subtract karna har configuration mein sahi cheez hai.

Ek-picture summary

Do overlapping blobs; annotation padhta hai area of A ∪ B = A + B − (the sliver). Woh single caption hi poori derivation hai.

Recall Feynman: ek story ki tarah batao

Duniya ko equal squares ki ek dari samjho — woh dari sab kuch jo ho sakta hai hai, aur uska total size 1 hai. Un squares ko colour karo jahan "event A" sach hai, aur alag se un squares ko colour karo jahan "event B" sach hai. Jaanna chahte ho A ya B kitna dari colour ki hai? Naively aap A ki amount aur B ki amount add karte. Lekin dekho — jo squares dono ke liye sach hain woh do baar colour ho gaye, toh aapne us chhote shared patch ke liye do baar pay kiya. Toh us shared patch ko ek baar subtract karo. Bas itna hi hai: A ya B = A plus B minus shared bit. Agar A aur B kabhi nahi milte, shared bit kuch nahi hai, aur aap bas add karo. Agar B, A ke andar chhupta hai, shared bit poora B hai, toh B add karne se kuch nahi badalta. Woh ek chhoti si subtraction har case fix karti hai.

Connections

Set Theory — Union, Intersection, Complement — $\cup, \cap, \emptyset$ algebra jo yeh tiles follow karte hain.
Conditional Probability & Bayes' Theorem — board ko ek single event mein zoom karo aur re-measure karo.
Independence of Events — jab sliver ka area $P (A) P (B)$ ke barabar hota hai.
Permutations & Combinations — pehle equal tiles ko count kaise karte hain.