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

2.7.8 · D1 · Maths › Statistics & Probability — Intermediate › Conditional probability — P(A - B) = P(A∩B) - P(A)

Yeh page parent topic mein use hone wale har mark ko scratch se build karti hai, us order mein jis order mein woh ek doosre par depend karte hain. Hum ek single dot se shuru karte hain aur, symbol by symbol, uss jagah khatam karte hain jahan parent ka formula kuch aisa ban jaata hai jise tum dekh sako, na ki sirf yaad karo. Neeche kuch bhi tab tak use nahi hota jab tak usse draw aur define na kar liya jaye.

0. Shuruwati picture: outcomes as dots

Sab kuch jo ho sakta hai uss se shuru hota hai. Ek experiment imagine karo — die roll karna, card draw karna. Jo bhi result ho sakta hai, woh ek outcome hai. Har outcome ko ek single dot ki tarah draw karo (figure s01).

Yahan se kyun shuru karein? Kyunki probability dots ginana hai. Agar tumhe pata nahi ek dot kya hai, tum unhe gin nahi sakte, aur baad ke har symbol secretly ek count hai.

1. Sample space $S$ — poori box

• Seedhe alfaz mein: har possible result, saath mein ikattha kiya hua.
• Picture: woh bahari rectangle jo saare dots ko pakde hua hai (figure s01).
• Topic ko kyun chahiye: conditional probability is box ko chhhota karne ke baare mein hai. Woh box jo tumne kabhi draw nahi ki, usse chhhota nahi kar sakte.

Sample space aur events ke poore treatment ke liye dekho Sample space and events। Hamare liye, $S$ bas "dots ki universe" hai।

2. Event — dots ka ek chuna hua circle

• Seedhe alfaz mein: result ke baare mein ek haan/naa ka sawaal, jaise "kya die ka score even hai?" Event un dots ka set hai jo haan ka jawaab dete hain.
• Picture: $S$ ke andar kuch dots ko gherta hua ek loop (figure s02).
• Topic ko kyun chahiye: parent mein name kiye gaye do events $A$ aur $B$ poori kahani hain. Sab kuch is baare mein hai ki dots ke do circles kaise overlap karte hain.

3. Counting symbol $n(\cdot )$ — kitne dots hain

• Seedhe alfaz mein: circle kitne dots pakad raha hai.
• Picture: loop ke andar har dot par apni ungli rakho aur gino.
• Topic ko kyun chahiye: parent ki derivation raw counts $n(B)$ aur $n(A\cap B)$ se shuru hoti hai, phir unhe probabilities mein convert karti hai.

4. Do loops saath: union $\cup$ aur overlap $\cap$

Jab do loops ek picture mein hoti hain, to bilkul do nayi ideas hoti hain — "koi bhi loop" aur "dono loops." Dono ko abhi naam dete hain taaki baad mein koi surprise na ho.

• Topic ko specifically $\cap$ kyun chahiye: conditional probability do-part ka sawaal poochhti hai — "kya $B$ hua, aur kya woh $A$ bhi tha?" Sirf overlap dono ka jawaab deta hai. Koi bhi akela loop nahi deta, aur union $\cup$ zyada weak "ya" waale sawaal ka jawaab deta hai.

5. Probability $P(\cdot )$ — box ka fraction

Ab hum counting ko probability mein convert karte hain. Yeh sabse important translation hai.

• Seedhe alfaz mein: sab tareekon mein se jis tarah cheezein ho sakti hain, us mein se kitna share $A$ ke andar aata hai?
• Picture: loop ko shade karo; $P(A)$ rectangle ka kitna hissa shade hua hai — $0$ aur $1$ ke beech ka ek number.
• Divide kyun karein, sirf count kyun nahi? Kyunki raw counts alag-alag experiments mein compare nahi kiye ja sakte. $n(S)$ se divide karne par sab kuch $0$-se-$1$ ruler par aa jaata hai, jahan $0=$ kabhi nahi aur $1=$ hamesha.

6. Bar "$\mid$" — matlab given

Yeh woh symbol hai jis par poora topic named hai. Isse pehle ki sab cheez earn ho chuki hai, to ab hum ise likh sakte hain.

• Seedhe alfaz mein: "$A$ ki probability, ab jab hum jaante hain ki $B$ sach hua."
• Picture: hum $B$ ke bahar ke har dot ko mita dete hain. Box literally sirf loop $B$ tak chhhota ho jaata hai — woh hamaari nayi universe ban jaati hai. Phir hum poochhte hain ki is chhhoti box ka kitna hissa $A$ bhi hai (figure s04).
• Topic ko kyun chahiye: yahi poori idea hai. Bar ke bina koi "kuch seekhna" nahi hai, aur probability kabhi update nahi hoti.

Kyunki nayi box $B$ hai, uski size — woh number jisse hume divide karna hai — count $n(B)$ hai। Figure s04 se seedha padh kar, chhhoti box ka woh fraction jo $A$ bhi hai: $P(A\mid B)=\frac{\text{overlap dots}}{\text{chhhoti box mein baaki dots}}=\frac{n(A\cap B)}{n(B)}.$

Isliye denominator $P(B)$ hai, $P(A)$ nahi: $B$ woh nayi (chhhoti) universe hai jiske andar hum count kar rahe hain.

7. Complement $A^c$ — woh sab jo $A$ nahi hai

Parent $A^c$ aur $B^c$ use karta hai (false-positive example aur complement law mein), to inhe bhi earn karte hain.

• Seedhe alfaz mein: "$A$ nahi hua."
• Picture: loop $A$ ko chhodkar poora rectangle shade kiya hua hai (figure s05).
• Topic ko kyun chahiye: true complement law $P(A\mid B)+P(A^c\mid B)=1$ kehta hai ki fixed box $B$ ke andar, ya tum $A$ mein ho ya nahi ho — shares box ko poora bharna chahiye.

8. Restriction $P(B)>0$ — box khali nahi ho sakti

• Picture: ek khali rectangle — ginne ke liye kuch nahi, condition karne ke liye kuch nahi.
• Topic ko kyun chahiye: yeh poore formula par ek aur keval ek legal fine-print hai.

Yeh topic ko kaise feed karte hain

Neeche diagram ek dependency map hai: ise upar ki taraf padho. Har box is page ka ek symbol hai; har arrow ka matlab hai "neeche waali idea pehle chahiye upar waali sense hone ke liye." Dots ek box banate hain, loops events banate hain, counting probability banati hai, aur bar box ko chhhota karta hai — jisse conditional probability nikalti hai.

Kaise padhen: agar tum neeche ke har box se "Conditional P of A given B" tak path trace kar sako aur har arrow zubaan se explain kar sako, to tum parent topic ke liye ready ho.

Equipment checklist

Khud test karo: daayein side cover karo aur reveal karne se pehle jawaab do. Agar koi fail ho, to woh section dobara padho.

Connections

• Sample space and events — woh box aur loops jo yahan build kiye gaye.
• Tree diagrams — boxes ko step by step chhhota hote dekhne ka ek aur tarika.
• Multiplication rule of probability — conditional formula ko rearrange karne par jo milta hai.
• Independence of events — woh special case jahan box ko chhhota karne se kuch nahi badalta.
• Bayes' Theorem — safely swap karna ki kaun sa event "given" hai.
• Law of total probability — box ke ek partition mein conditionals ko jodna.