Complementary events — P(A') = 1 − P(A)

1.3.7 · Maths › Basic Data & Probability

What is a Complementary Event?

Key property: A aur A' hote hain:

1. Mutually exclusive: $A\cap A^{\prime }=\varnothing$ (inme koi bhi outcome common nahi)
2. Exhaustive: $A\cup A^{\prime }=S$ (milake poora sample space cover karte hain)

Derivation: P(A') = 1 − P(A) kyun hota hai

Chalo ise probability ke axioms se scratch se banate hain.

Step 1: Sample space ko partition karo Kyunki A aur A' exhaustive hain: $A\cup A^{\prime }=S$

Step 2: Addition rule lagao Kyunki A aur A' mutually exclusive hain ($A\cap A^{\prime }=\varnothing$): $P(A\cup A^{\prime })=P(A)+P(A^{\prime })$

Kyun? Addition rule kehta hai: agar do events ek saath nahi ho sakte, toh "koi bhi ek" ki probability unki individual probabilities ka sum hoti hai.

Step 3: Total probability axiom use karo Poore sample space ki probability 1 hoti hai: $P(S)=1$

Step 4: Combine karo Step 1 se: $A\cup A^{\prime }=S$, toh $P(A\cup A^{\prime })=P(S)=1$

Step 2 se: $P(A\cup A^{\prime })=P(A)+P(A^{\prime })$

Isliye: $P(A)+P(A^{\prime })=1$

Ye hamesha true hota hai kisi bhi probability space mein kisi bhi event A ke liye.

Complement kab use karein

Worked Examples

Direct approach (lamba):

• Favorable outcomes: {1, 2, 3, 4, 5} → 5 outcomes
• P(not 6) = 5/6

Complement approach (faster):

• Maano A = "6 aana"
• P(A) = 1/6
• P(A') = 1 − 1/6 = 5/6 ✓

Ye step kyun? Hum pehle se jaante hain P(6) simple hai; 1 mein se ghatana paanch outcomes count karne ki jagah ek hi calculation hai.

Direct approach (tedious):

• Saare outcomes list karo jisme ≥1 head ho: HHH, HHT, HTH, HTT, THH, THT, TTH → 7 outcomes
• Sample space: 2³ = 8
• P(at least 1 H) = 7/8

Complement approach (elegant):

• Maano A = "kam se kam ek head"
• A' = "koi head nahi" = "sab tails" = {TTT}
• P(A') = 1/8
• P(A) = 1 − 1/8 = 7/8 ✓

Ye kyun kaam karta hai? "At least one" mein 7 cases hain; "none" mein sirf 1 case hai. Complement problem ko chhota kar deta hai.

Setup:

• Maano A = "spade draw karna"
• P(A) = 13/52 = 1/4
• P(A') = 1 − 1/4 = 3/4

Ye step kyun? Hum jaante hain spades = 13 cards turant. Hearts + diamonds + clubs = 39 cards count karna zyada kaam hai.

Solution:

• Maano A = "defect pakda gaya"
• P(A) = 0.95
• P(A') = 1 − 0.95 = 0.05 (5% nikal jaate hain)

Ye kyun important hai? Reliability engineering mein, failure rate (complement) jaanna aksar success rate se zyada critical hota hai.

Common Mistakes

Kyun sahi lagta hai: Hum sochte hain "not rain = sunny"

Fix: Not rain mein cloudy, snowy, foggy, etc. bhi ho sakta hai. Complement hai "koi bhi weather jo rain nahi ho." Complement rule tab hi use karo jab A' ko precisely "A ke siwa sab kuch" ke roop mein define kiya gaya ho.

P(rain') = P(not rain) = 0.7 mein saara non-rain weather shamil hai, sirf sunny nahi.

Kyun sahi lagta hai: Ye sach hai ki P(A) + P(A') = 1, isliye hum is sum ko use karna chahte hain.

Fix: A aur A' by definition mutually exclusive aur exhaustive hain. Tum inhe is logic se doosre events ke saath meaningfully combine nahi kar sakte. Equation P(A) + P(A') = 1 ek identity hai, doosre events ke liye calculation tool nahi.

Agar tumhare paas events B aur C hain, tum P(B ∪ C) = P(B) + P(C) nahi keh sakte jab tak pehle verify na karo ki wo mutually exclusive hain.

Kyun hota hai: Hum bhool jaate hain ki subtraction aksar re-counting se faster hoti hai.

Fix: Hamesha poochho: "Kya mujhe pehle se P(A) pata hai? Kya complement aasaan hai?" Agar haan, toh 1 − P(A) use karo.

Tum count kar sakte ho: 10 mein se 7 neeli marbles → 7/10.

Lekin shortcut ye hai: Agar laal ki chance 3/10 hai, toh "not red" ki chance wo hai jo 100% mein se bachi hai. Toh: 1 − 3/10 = 7/10.

Ye kyun kaam karta hai? Kyunki har marble ya toh laal hai ya nahi. Koi teesra option nahi. Toh laal + not-red = saari marbles = 100% = 1.

Ye trick powerful hai: Agar tum kisi cheez ka "at least one" dhundhne ki koshish kar rahe ho (jaise "3 coin flips mein kam se kam ek head"), toh "zero" (sab tails) calculate karna aur 1 mein se ghatana bahut aasaan hai. 7 cases count karne ki jagah, sirf 1 case count karo!

Ya: "Milke, wo ONE banate hain"

• A aur A' milke = poora sample space = probability 1

Connections

• 1.3.01-Sample-space-and-events — A' sample space S ka ek subset hai
• 1.3.03-Addition-rule-for-probabilities — P(A) + P(A') = 1 prove karne ke liye use kiya
• 1.3.09-Independent-events — P(A' ∩ B) = P(A') · P(B) agar independent hon
• 1.3.12-At-least-one-problems — Complements ka killer application
• 1.4.05-Binomial-probability — P(X ≥ 1) = 1 − P(X = 0) complements use karta hai

#flashcards/maths

Event A ka complement kya hota hai? :: Sample space ke un saare outcomes ka set jo A mein NAHI hain, jise A' ya A^c likha jaata hai. Ye represent karta hai "A nahi hota."

Complement rule ka formula batao :: P(A') = 1 − P(A), jo iss baat se aata hai ki A aur A' mutually exclusive aur exhaustive hote hain.

A aur A' mutually exclusive kyun hote hain? :: Kyunki unme koi outcome common nahi hota: A ∩ A' = ∅. Koi bhi outcome ek saath "A mein" aur "A mein nahi" nahi ho sakta.

True ya False: P(A) + P(A') hamesha 1 ke barabar hota hai :: True, kisi bhi probability space mein kisi bhi event A ke liye, kyunki A aur A' sample space ko partition karte hain.

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