Mutually exclusive events — addition rule — Hinglish notes

2.7.6 · Maths › Statistics & Probability — Intermediate

WHAT — hum kis baare mein baat kar rahe hain?

WHY yeh matter karta hai: "2 roll karo" aur "5 roll karo" ek hi die par dono ek saath nahi ho sakte — mutually exclusive. Lekin "even number roll karo" aur " $3$ se zyada number roll karo" dono ho sakte hain (4 ya 6) — not mutually exclusive.

Figure — Mutually exclusive events — addition rule

HOW — addition rule ko first principles se derive karte hain

Hum general rule se shuru karte hain aur special case earn karte hain.

Step 1 — Outcomes count karo. Equally likely outcomes ke liye, $P (E) = \frac{n ( E )}{n ( S )}$ jahan $n (\cdot)$ outcomes ko count karta hai.

Yeh step kyun? Kisi event ki probability = (favourable outcomes) ÷ (total outcomes). Sab kuch honest counting se aata hai.

Step 2 — $A \cup B$ count karo. " $A$ ya $B$ " mein kitne outcomes hain? Agar hum naively $n (A) + n (B)$ add karein, toh jo bhi outcome dono $A$ aur $B$ mein hai woh do baar count hoga. Double count fix karne ke liye, overlap ko ek baar subtract karo: $n (A \cup B) = n (A) + n (B) - n (A \cap B)$

Yeh step kyun? Yeh inclusion–exclusion idea hai: pieces add karo, phir jo double add hua use hata do.

Step 3 — $n (S)$ se divide karo. $\frac{n ( A \cup B )}{n ( S )} = \frac{n ( A )}{n ( S )} + \frac{n ( B )}{n ( S )} - \frac{n ( A \cap B )}{n ( S )}$ jo hai general addition rule: $P (A \cup B) = P (A) + P (B) - P (A \cap B)$

Step 4 — Mutual exclusivity apply karo. Agar $A$ aur $B$ mutually exclusive hain, toh $A \cap B = \emptyset$ , isliye $P (A \cap B) = 0$ . Correction term gayab ho jaata hai:

Handy corollary (complement): Kyunki $A$ aur uska complement $A^{'}$ mutually exclusive hain aur saath mein $S$ ko poora fill karte hain: $P (A) + P (A^{'}) = 1 \Rightarrow P (A^{'}) = 1 - P (A)$

Worked examples

Forecast-then-Verify

Recall Pehle predict karo, phir reveal karo

Q: Do events jisme $P (A) = 0.6$ , $P (B) = 0.7$ . Kya woh mutually exclusive ho sakte hain? Forecast: … Verify: Agar exclusive hain, toh $P (A \cup B) = 0.6 + 0.7 = 1.3 > 1$ — impossible (probabilities max 1 tak hoti hain). Isliye woh mutually exclusive nahi ho sakte; unka overlap kam se kam $0.6 + 0.7 - 1 = 0.3$ toh hoga hi.

Common mistakes (Steel-man karte hain)

80/20 — ek cheez jo yaad rakhni hai

Recall Feynman: ek 12-saal ke bacche ko samjhao

Socho toys ke boxes hain. "Mutually exclusive" ka matlab hai koi toy sirf ek box mein reh sakti hai — koi toy do boxes mein ek saath nahi hai. Agar main poochhuun "kya chance hai ki main box A ya box B se toy uthaaunga?", toh main bas dono boxes kitne bhari hain woh add kar dunga, kyunki koi toy do baar count nahi hua. Lekin agar kuch toys dono boxes mein hoti, toh maine unhe do baar count kar liya hota, isliye unhe ek baar nikalna padta. Jab boxes mein kuch common nahi hai, toh nikalne ko kuch nahi — isliye seedha add karna kaam karta hai!

Active-recall flashcards

Do events mutually exclusive kab hoti hain…

jab unka intersection empty ho,

A \cap B = \emptyset

(dono ek saath nahi ho sakti).

General addition rule batao.

P (A \cup B) = P (A) + P (B) - P (A \cap B)

Mutually exclusive events ke liye addition rule batao.

P (A \cup B) = P (A) + P (B)

General rule mein

P (A \cap B)

kyun subtract karte hain?

Dono events mein jo outcomes hain woh add karte waqt do baar count hote hain; ek baar subtract karna double count theek karta hai.

Kya mutually exclusive events independent hoti hain?

Nahi — nonzero probabilities ke liye woh strongly dependent hoti hain (agar ek ho gayi toh doosri nahi ho sakti).

P (A) = 0.6, P (B) = 0.7

— kya woh mutually exclusive ho sakte hain?

Nahi, kyunki

0.6 + 0.7 = 1.3 > 1

impossible hai.

Addition rule se complement rule?

P (A^{'}) = 1 - P (A)

, kyunki

A, A^{'}

exclusive aur exhaustive hain.

Ek die par,

P (even or > 3)

\frac{3}{6} + \frac{3}{6} - \frac{2}{6} = \frac{2}{3}

(simply

1

nahi).

Add vs multiply — quick cue?

OR ke liye add (alternatives), AND ke liye multiply (independent, meanwhile).

Connections

Probability — basic definitions & sample space
General addition rule & inclusion–exclusion
Independent events — multiplication rule
Complementary events
Conditional probability
Venn diagrams in probability