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Mutually Exclusive and Exhaustive Events

Introduction

In Probability में events एक-दूसरे से अलग-अलग relationships रख सकते हैं। इनमें से दो सबसे महत्वपूर्ण relationships हैं Mutually Exclusive और Exhaustive events। इनको समझने से आप यह analyze कर पाते हैं कि किसी experiment में outcomes आपस में कैसे interact करते हैं।

ये concepts खास तौर पर तब उपयोग होते हैं जब probabilities को add करना हो या combined events को समझना हो (जैसे “either A या B होता है”).

Pattern: Mutually Exclusive and Exhaustive Events

Pattern

Mutually Exclusive events एक साथ नहीं हो सकते, जबकि Exhaustive events सभी possible outcomes को cover करते हैं।

  • Mutually Exclusive: P(A ∩ B) = 0
  • Exhaustive Events: P(A) + P(B) + ... = 1

Example: Coin toss करने पर Head मिलना और Tail मिलना mutually exclusive हैं (दोनों साथ नहीं हो सकते), और दोनों मिलकर exhaustive भी हैं क्योंकि इनमें से एक outcome ज़रूर होगा।

Step-by-Step Example

Question

एक die roll करने पर event A = “even number मिलना” और event B = “odd number मिलना” हैं। क्या ये events mutually exclusive और/या exhaustive हैं?

Solution

  1. Step 1: Outcomes define करें

    A = {2, 4, 6} और B = {1, 3, 5}.

  2. Step 2: Intersection check करें

    A ∩ B = {} → empty set, इसलिए ये mutually exclusive हैं।

  3. Step 3: Exhaustiveness check करें

    A ∪ B = {1, 2, 3, 4, 5, 6} → die के सभी outcomes → exhaustive।

  4. Final Answer:

    mutually exclusive और exhaustive।

  5. Quick Check:

    P(A ∩ B) = 0 और P(A ∪ B) = 1 ✅

Quick Variations

1. दो cards निकालने पर “red” और “black” suits → mutually exclusive और exhaustive।

2. Even number vs multiple of 3 on a die → mutually exclusive नहीं (क्योंकि 6 common है)।

3. Die roll कर A = {1,2,3}, B = {4,5,6} → exhaustive और mutually exclusive।

Trick to Always Use

  • Step 1: Intersection देखें - अगर A ∩ B = ∅ है, तो events mutually exclusive हैं।
  • Step 2: Union देखें - अगर A ∪ B सभी outcomes cover करता है, तो events exhaustive हैं।
  • Step 3: Mutually exclusive events के लिए formula: P(A ∪ B) = P(A) + P(B).

Summary

Summary

Mutually Exclusive and Exhaustive Events pattern में:

  • Mutually exclusive → एक साथ occur नहीं होंगे (P(A ∩ B) = 0).
  • Exhaustive → इनमें से कोई न कोई outcome ज़रूर होगा (sum of probabilities = 1).
  • ये concepts “either-or” वाले probability problems को सरल बनाते हैं।
  • Events classify करने के लिए हमेशा intersection और union दोनों check करें।

Practice

(1/5)
1. When a fair coin is tossed once, let event A = {Head} and event B = {Tail}. Which statement is true?
easy
A. They are both mutually exclusive and exhaustive
B. They are mutually exclusive only
C. They are exhaustive only
D. They are neither

Solution

  1. Step 1: List outcomes

    Sample space S = {H, T}.

  2. Step 2: Check intersection

    A ∩ B = ∅ → cannot occur together → mutually exclusive.

  3. Step 3: Check union

    A ∪ B = {H, T} = S → covers all outcomes → exhaustive.

  4. Final Answer:

    mutually exclusive and exhaustive → Option A.

  5. Quick Check:

    P(A ∩ B) = 0 and P(A ∪ B) = 1 ✅
Hint: Mutually exclusive → intersection empty. Exhaustive → union = sample space.
Common Mistakes: Thinking 'mutually exclusive' alone implies not exhaustive.
2. On a single roll of a fair six-sided die, let A = 'even number' and B = 'multiple of 3'. Which classification is correct?
easy
A. Both mutually exclusive and exhaustive
B. Mutually exclusive only
C. Exhaustive only
D. Neither mutually exclusive nor exhaustive

Solution

  1. Step 1: List sets

    A = {2,4,6}, B = {3,6}.

  2. Step 2: Check intersection

    A ∩ B = {6} ≠ ∅ → not mutually exclusive.

  3. Step 3: Check union

    A ∪ B = {2,3,4,6} which ≠ {1,2,3,4,5,6} → not exhaustive.

  4. Final Answer:

    Neither mutually exclusive nor exhaustive → Option D.

  5. Quick Check:

    6 belongs to both (so not exclusive); 1 and 5 are missing from union (so not exhaustive) ✅
Hint: Look for shared elements (intersection) first; then check if union covers all outcomes.
Common Mistakes: Assuming even vs multiple-of-3 are disjoint or cover the whole sample space.
3. Events A = {1,2,3} and B = {4,5} are defined on a single roll of a fair six-sided die. Which statement is correct?
easy
A. Both mutually exclusive and exhaustive
B. Mutually exclusive only
C. Exhaustive only
D. Neither

Solution

  1. Step 1: Identify sets

    A = {1,2,3}, B = {4,5}.

  2. Step 2: Intersection

    A ∩ B = ∅ → they are mutually exclusive.

  3. Step 3: Union

    A ∪ B = {1,2,3,4,5} which does not include 6 → not exhaustive.

    4. Final Answer:

    Mutually exclusive only → Option B.

  4. Quick Check:

    They don't overlap, but their union misses outcome 6 → not exhaustive ✅
Hint: Mutual exclusion is intersection-empty; exhaustiveness needs full coverage of sample space.
Common Mistakes: Confusing non-overlap with covering all outcomes.
4. If events A and B are mutually exclusive and P(A) = 0.30, P(B) = 0.40, what is P(A ∪ B)?
medium
A. 0.10
B. 0.12
C. 0.70
D. 0.18

Solution

  1. Step 1: Use formula for mutually exclusive events

    If A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B).

  2. Step 2: Substitute values

    P(A ∪ B) = 0.30 + 0.40 = 0.70.

  3. Final Answer:

    0.70 → Option C.

  4. Quick Check:

    Since they can't occur together, union is sum of probabilities → 0.70 ✅
Hint: For mutually exclusive events, simply add probabilities for the union.
Common Mistakes: Trying to subtract intersection (which is zero here) or multiplying instead of adding.
5. Let event A = 'draw a red card' and event B = 'draw a black card' from a standard 52-card deck. Which classification fits A and B?
medium
A. They are both mutually exclusive and exhaustive
B. Mutually exclusive only
C. Exhaustive only
D. Neither

Solution

  1. Step 1: Identify sets

    Red cards = 26 cards, Black cards = 26 cards.

  2. Step 2: Intersection

    Red ∩ Black = ∅ → mutually exclusive.

    3. Step 3: Union

    Red ∪ Black = all 52 cards → exhaustive.

  3. Final Answer:

    Both mutually exclusive and exhaustive → Option A.

  4. Quick Check:

    P(Red ∩ Black)=0 and P(Red ∪ Black)=1 ✅
Hint: Opposite exhaustive categories (like red/black) are usually both exclusive and exhaustive.
Common Mistakes: Thinking suits (Hearts/Spades) are exhaustive of colours (they're not).

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