Foundations — Independent events — multiplication rule

2.7.7 · D1 · Maths › Statistics & Probability — Intermediate › Independent events — multiplication rule

Yeh page toolbox hai. Parent note (Independent Events) mein $P(A)$, $A\cap B$, $P(A\mid B)$, aur $\prod$ jaise symbols use hote hain. Us note ko chhune se pehle, tumhe inme se har ek dikhna chahiye. Hum har ek ko zero se banate hain, us order mein jisme yeh ek doosre par depend karte hain.

1. Ek outcome, ek event, aur sample space

Sab kuch ek random experiment se shuru hota hai: coin flip karo, die roll karo, marble draw karo.

Sample space ko ek box ke roop mein socho jo har possible outcome ko hold karta hai. Ek event ek region hai jo us box ke andar draw ki gayi hai — un outcomes ke around ek fence jo "haan" mein count hoti hain.

Topic ko yeh kyun chahiye: tum "$A$ aur $B$ dono hote hain" nahi keh sakte jab tak $A$ aur $B$ ek hi box mein regions na hon. Poora multiplication rule isi baare mein hai ki do aisi regions kaise overlap karti hain.

2. Symbol $P()$ — probability as a fraction of the box

Topic ko yeh kyun chahiye: multiplication rule inhi fractions ko multiply karta hai. Agar tum $P(A)$ ko "kitna box" feel nahi karte, toh product $P(A)P(B)$ sirf meaningless symbols hain.

3. Do events ek saath: $\cap$ (AND) aur $\cup$ (OR)

Ab ek hi box mein do fenced regions, $A$ aur $B$, rakh do. Yeh overlap kar sakti hain.

Topic ko yeh kyun chahiye: poora rule $P(A\cap B)$ ke baare mein hai — overlap ki probability. "$P(\text{both heads})$" hai hi $P(H_1\cap H_2)$. Overlap picture galat samjho toh har example collapse ho jaata hai.

4. Conditional probability $P(A\mid B)$ — box ko chhhota karna

Yeh sabse important symbol hai jo internalize karna hai, kyunki poora rule isi se derive hota hai.

Figure dekho: accent-red slice $A\cap B$ hai. Conditioning se pehle hum ise poore box se compare karte the; conditioning ke baad hum ise sirf $B$ (shaded region) se compare karte hain. Same slice, bada fraction, kyunki denominator chhhota ho gaya.

Topic ko yeh kyun chahiye: independence isi symbol se define hoti hai. Do events independent hain tab jab $B$ jaanna cheezein shrink-and-reshape nahi karta — yaani $P(A\mid B)=P(A)$. Parent note upar boxed formula ko rearrange karke multiplication rule banata hai, toh tumhare paas yeh hona zaroori hai.

5. Independence ka idea ek picture mein

Topic ko yeh kyun chahiye: yahi destination hai. Is page ki baaki sab cheez is liye exist karti hai taaki yeh line obvious lage, magical nahi.

6. Complement $A^c$ aur $1-P(A)$

Topic ko yeh kyun chahiye: "at least one" trick. "Kam se kam ek cheez hoti hai" directly count karna mushkil hai, lekin uska complement — "koi nahi hota" — ek clean product hai. Toh topic likhta hai $P(\text{at least one})=1-P(\text{none}).$

7. Product symbol $\prod$ — repeated multiplication

Topic ko yeh kyun chahiye: kai independent events ke liye parent likhta hai $P(A_1\cap \cdots \cap A_n)={\prod }_{i}P(A_i)$ aur $P(\text{at least one})=1-{\prod }_i(1-P(A_i))$. $\prod$ ke bina yeh endless "$\cdots$" chains hote.

Yeh foundations topic ko kaise feed karte hain

Equipment checklist

Cover the right side and answer out loud before opening the parent note.

Connections

• Conditional Probability — Section 4 woh seed hai jis se parent ki derivation grow hoti hai.
• Complement Rule — Section 6, "at least one" shortcut ko power karta hai.
• Mutually Exclusive Events — overlap $A\cap B$ ka empty hona; independence se contrast.
• General Multiplication Rule — Section 4 kya banta hai independence impose hone se pehle.
• Bayes' Theorem — conditional-probability box-shrinking idea par bhi bana hai.
• Binomial Distribution — Section 7 ke $\prod$ ka use karke kai independent trials stack karta hai.