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Complementary Events

Introduction

Probability में, हर event का एक complementary event होता है - ऐसा event जो उन सभी outcomes को दिखाता है जिनमें original event नहीं होता। Complementary events को समझना बहुत जरूरी है, खासकर जब किसी समस्या में यह calculate करना आसान हो कि क्या नहीं हुआ, बजाय इसके कि क्या हुआ।

यह pattern probability calculations को आसान बनाता है एक basic rule का उपयोग करके: P(not E) = 1 - P(E).

Pattern: Complementary Events

Pattern

किसी event के न होने की probability = 1 - उस event के होने की probability।

Formula: P(not E) = 1 - P(E)

क्योंकि total probability हमेशा 1 होती है, इसलिए यह relationship एक को जानते ही दूसरी को जल्दी calculate करने में मदद करता है।

Step-by-Step Example

Question

एक die को एक बार roll किया जाता है। 6 न आने की probability निकालो।

Solution

  1. Step 1: सभी possible outcomes लिखें

    Die पर total possible outcomes = {1, 2, 3, 4, 5, 6} → total = 6.

  2. Step 2: Favourable outcomes पहचानें

    6 आने के favourable outcomes = 1 (सिर्फ face ‘6’).

  3. Step 3: 6 आने की probability निकालें

    P(getting a 6) = 1 / 6 = 1/6.

  4. Step 4: Complementary rule लागू करें

    P(not getting a 6) = 1 - 1/6 = 5/6.

  5. Final Answer:

    5/6

  6. Quick Check:

    P(getting 6) + P(not getting 6) = 1/6 + 5/6 = 1 ✅

Quick Variations

1. “At least one” वाले problems (जैसे, P(at least one head) = 1 - P(no heads)).

2. किसी खास card या color के न आने की probability निकालना।

3. “none”, “not happening”, या “failure” outcomes वाली situations।

Trick to Always Use

  • Step 1: Event के होने की probability P(E) निकालें।
  • Step 2: 1 में से subtract करें और P(not E) पाएँ।
  • Step 3: हमेशा verify करें कि P(E) + P(not E) = 1.

Summary

Summary

Complementary Events pattern में:

  • P(not E) = 1 - P(E).
  • जब opposite event निकालना आसान हो, तब इसका उपयोग करें।
  • Total probability = 1 को हमेशा verify करें।
  • “at least one” और “none” type problems में बहुत common है।

Practice

(1/5)
1. A box contains 7 white balls and 3 black balls. One ball is drawn at random. What is the probability that the ball is not black?
easy
A. 7/10
B. 3/10
C. 1/2
D. 4/5

Solution

  1. Step 1: Identify total outcomes

    Total balls = 7 white + 3 black = 10.

  2. Step 2: Identify P(black)

    P(black) = 3 / 10 = 3/10.

  3. Step 3: Use complement

    P(not black) = 1 - P(black) = 1 - 3/10 = 7/10.

  4. Final Answer:

    7/10 → Option A.

  5. Quick Check:

    Not black = white count 7 → 7/10 ✅
Hint: Find P(E) for the unwanted event first, then subtract from 1.
Common Mistakes: Forgetting to include all balls when computing the total.
2. Two fair coins are tossed. What is the probability of getting at least one Head?
easy
A. 3/4
B. 1/4
C. 1/2
D. 1

Solution

  1. Step 1: Identify total outcomes

    Sample space for two coins = {HH, HT, TH, TT} → 4 outcomes.

  2. Step 2: Identify complement (no Heads)

    Complement = {TT} → P(no Heads) = 1/4.

  3. Step 3: Apply complement rule

    P(at least one Head) = 1 - P(no Heads) = 1 - 1/4 = 3/4.

  4. Final Answer:

    3/4 → Option A.

  5. Quick Check:

    Direct count: {HH, HT, TH} = 3 favourable out of 4 → 3/4 ✅
Hint: For 'at least one' problems, compute P(none) and subtract from 1.
Common Mistakes: Counting HT and TH as one outcome (they are distinct) leading to wrong totals.
3. One card is drawn from a standard 52-card deck. What is the probability that the card is not a Spade?
easy
A. 1/4
B. 3/4
C. 1/2
D. 13/52

Solution

  1. Step 1: Identify total outcomes

    Total cards = 52.

  2. Step 2: Identify P(Spade)

    Number of spades = 13 → P(Spade) = 13/52 = 1/4.

  3. Step 3: Use complement

    P(not Spade) = 1 - P(Spade) = 1 - 1/4 = 3/4.

  4. Final Answer:

    3/4 → Option B.

  5. Quick Check:

    Spades = 13, not spade = 39 → 39/52 = 3/4 ✅
Hint: P(not E) = 1 - P(E); for suits P(E) often equals 13/52 = 1/4.
Common Mistakes: Using 26 (red cards) values instead of correctly identifying the suit count.
4. A bag contains 4 red and 6 blue balls. One ball is drawn at random. What is the probability that the drawn ball is not blue?
medium
A. 3/5
B. 7/10
C. 2/5
D. 4/6

Solution

  1. Step 1: Identify total outcomes

    Total balls = 4 red + 6 blue = 10.

  2. Step 2: Identify P(blue)

    P(blue) = 6 / 10 = 3/5.

  3. Step 3: Use complement

    P(not blue) = 1 - P(blue) = 1 - 3/5 = 2/5.

  4. Final Answer:

    2/5 → Option C.

  5. Quick Check:

    Not blue = red count 4 → 4/10 = 2/5 ✅
Hint: Either compute 1 - P(blue) or directly use favourable count for 'not' event.
Common Mistakes: Forgetting to compute total before applying complement.
5. Three fair coins are tossed. What is the probability that the outcome is not exactly one Head?
medium
A. 3/8
B. 1/2
C. 5/16
D. 5/8

Solution

  1. Step 1: Identify total outcomes

    Total outcomes for 3 coins = 2³ = 8.

  2. Step 2: Identify P(exactly one Head)

    Exactly one Head outcomes = {HTT, THT, TTH} → 3 outcomes → P(exactly one Head) = 3/8.

  3. Step 3: Use complement

    P(not exactly one Head) = 1 - 3/8 = 5/8.

  4. Final Answer:

    5/8 → Option D.

  5. Quick Check:

    Other outcomes count = 8 - 3 = 5 → 5/8 ✅
Hint: Compute the unwanted case count (exactly one) and subtract from 1.
Common Mistakes: Miscounting arrangements for exactly one Head (forgetting permutations).

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