Independent events — multiplication rule — Hinglish notes

2.7.7 · Maths › Statistics & Probability — Intermediate

HUM KISKE BAARE MEIN BAAT KAR RAHE HAIN?

Upar ke dono boxes ek hi baat keh rahe hain. Pehla matlab hai (koi information transfer nahi hoti); doosra multiplication rule hai jisse tum calculate karte ho.

MULTIPLY KYUN KAAM KARTA HAI? (Scratch se derive karo)

Hum multiplication rule ko kabhi bhi sirf "state" nahi karte — hum use conditional probability ki definition se build karte hain.

Toh multiplication rule koi nayi law nahi hai — yeh general rule plus independence assumption hai. Bas, yahi poori kahaani hai.

Figure — Independent events — multiplication rule

Bahut saare events tak extend karna

Ek kaam ki consequence — complement se "at least one": $P (at least one occurs) = 1 - \prod_{i} (1 - P (A_{i}))$ Kyun? "At least one" ka opposite hai "koi nahi hota." Sab fail ek saath hote hain probability $\prod (1 - P (A_{i}))$ se (failures ki independence use karke), toh 1 mein se subtract karo.

Worked examples

Apni galtiyon ko steel-man karo

Recall Feynman: ise ek 12 saal ke bacche ko samjhao

Socho tum ek saath coin flip kar rahe ho aur die roll kar rahe ho. Coin ko koi idea nahi ki die kya kar raha hai — woh ek doosre se baat nahi karte. Toh "heads AND six" ki chance ke liye, heads ki chance ( $\frac{1}{2}$ ) aur six ki chance ( $\frac{1}{6}$ ) lo aur multiply karo: $\frac{1}{12}$ . Multiply karna aisa hai jaise kehna "heads ke aadhe times mein... aur six ke chhathe hisse ke times mein." Lekin agar red marble nikalte hi ek red marble kam ho jaata hai, toh marbles baat karte hain — doosra pull pehle ke baare mein jaanta hai — toh tum wahi numbers simply multiply nahi kar sakte; tumhe update karna hoga.

Flashcards

What is the multiplication rule for independent events?

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

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

use karne ke liye kaunsi condition hold karni chahiye?

A

aur

B

independent hone chahiye, yani

P (A ∣ B) = P (A)

Independence multiplication rule ko conditional probability se derive karo.

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

; independence deta hai

P (A ∣ B) = P (A)

, toh

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

Kya mutually exclusive events (dono positive prob ke saath) independent hain?

Nahi — woh strongly dependent hain, kyunki

P (A \cap B) = 0 \neq = P (A) P (B)

P (at least one of independent A_{i})

ka formula?

1 - \prod_{i} (1 - P (A_{i}))

Bag: 4 red, 6 blue, replacement KE SAATH do draws, P(both red)?

\frac{4}{10} \cdot \frac{4}{10} = 0.16

Wahi bag REPLACEMENT KE BINA, P(both red)?

\frac{4}{10} \cdot \frac{3}{9} = \frac{2}{15} \approx 0.133

Teen parts har ek prob 0.9 se independently kaam karte hain; P(at least one fails)?

1 - 0. 9^{3} = 0.271

P (A ∣ B) = P (A)

ka matlab words mein kya hai?

B

ke hone ka pata chalna

A

ki probability nahi badalta.

Connections

Conditional Probability — multiplication rule directly isi se derive hota hai.
General Multiplication Rule — $P (A \cap B) = P (A) P (B ∣ A)$ , dependent case ka parent.
Mutually Exclusive Events — contrast: exclusive ≠ independent.
Complement Rule — "at least one" trick ko power karta hai.
Bayes' Theorem — yeh bhi conditional probability par bana hai.
Binomial Distribution — baar baar independent trials multiply hote hain.