Conditional probability — P(A - B) = P(A∩B) - P(A)

2.7.8 · Maths › Statistics & Probability — Intermediate

WHAT define kiya ja raha hai

$A \cap B$ = " $A$ aur $B$ dono" (overlap).
Hum ==universe ko $B$ tak restrict karte hain== aur dekhte hain ki $B$ ka kitna hissa $A$ le raha hai.

WHY formula aisa hi hona chahiye (derivation scratch se)

Maan lo outcomes equally likely hain, total $N$ outcomes hain.

$P (B) = \frac{n ( B )}{N}$ jahaan $n (B)$ = $B$ mein outcomes ki sankhya.
Jaise hi hum jaante hain ki $B$ hua, sirf $B$ ke $n (B)$ outcomes hi possible reh jaate hain. Nayi sample space hai hi $B$ .
Inme se, jo $A$ bhi dete hain unki sankhya $n (A \cap B)$ hai.

Toh chhhoti duniya ke andar basic "favourable / total" rule se: $P (A ∣ B) = \frac{n ( A \cap B )}{n ( B )}$

Ab upar aur neeche dono ko $N$ se divide karo (yeh trick "equally likely" assumption hataa deti hai): $P (A ∣ B) = \frac{n ( A \cap B ) / N}{n ( B ) / N} = \frac{P ( A \cap B )}{P ( B )}$

Figure — Conditional probability — P(A - B) = P(A∩B) - P(A)

Multiplication rule (rearranged form)

Dono sides ko $P (B)$ se multiply karo: $P (A \cap B) = P (B) P (A ∣ B) = P (A) P (B ∣ A)$

Agar $B$ jaanna $A$ ke baare mein kuch nahi batata, toh $P (A ∣ B) = P (A)$ . Substitute karo: $P (A \cap B) = P (B) P (A ∣ B) = P (B) P (A)$ yeh independence condition hai. Isliye: $A, B independent ⟺ P (A ∣ B) = P (A) ⟺ P (A \cap B) = P (A) P (B)$

Worked examples

Common mistakes (Steel-manned)

Flashcards

P (A ∣ B)

ko formula ke roop mein define karo (restriction bhi batao).

P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}

, valid for

P (B) > 0

Formula mein

P (A)

se nahi,

P (B)

se kyun divide karte hain?

Kyunki

B

given event hai — nayi sample space;

P (B)

se rescaling se

B

ke andar probabilities ka sum 1 ho jaata hai.

Multiplication rule do tarikhon se batao.

P (A \cap B) = P (A) P (B ∣ A) = P (B) P (A ∣ B)

Conditional probability use karke

A, B

independent hone ki condition.

P (A ∣ B) = P (A)

(equivalently

P (A \cap B) = P (A) P (B)

Kya

P (A ∣ B) = P (B ∣ A)

generally hota hai?

Nahi — numerator

P (A \cap B)

same hai lekin denominators alag hain.

Kaun si complement identity SAHI hai?

P (A ∣ B) + P (A^{c} ∣ B) = 1

(condition

B

change nahi kar sakte).

Fair die:

P (> 3 ∣ even)

{4, 6}

out of

{2, 4, 6}

2/3

P (two kings, no replacement)

from 52 cards?

\frac{4}{52} \cdot \frac{3}{51} = \frac{1}{221}

Recall Feynman: 12-saal ke bacche ko samjhao

Socho ek bade dabbe mein rang-birange marbles hain. Normally tum poore dabbe se ek red marble milne ki chance guess karte ho. Ab ek dost kehta hai: "Main tumhe sirf blue-lid waale hisse mein haath daalne deta hoon." Ek dam se tumhara dabba chhota ho gaya! Conditional probability ka bas yahi matlab hai: given ki tum chhhote (blue-lid waale) hisse mein ho, marble ke red hone ki kya chance hai? Tum usi hisse mein reds ginoge, aur us hisse ke total marbles se divide karoge. Woh "us hisse mein kitne marbles hain" hi $P (B)$ hai jo neeche aata hai.

Connections

Independence of events — special case jahaan $P (A ∣ B) = P (A)$ .
Multiplication rule of probability — rearranged form.
Bayes' Theorem — $P (A ∣ B)$ ko sahi tarike se $P (B ∣ A)$ mein flip karta hai.
Law of total probability — ek partition par conditionals ka sum.
Sample space and events — woh "universe" jo hum chhhota karte hain.
Tree diagrams — conditional steps ki visual chaining.