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Mutually Exclusive and Exhaustive Events

Introduction

In probability, events can have different types of relationships with each other. Two of the most important relationships are Mutually Exclusive and Exhaustive events. Understanding these helps to analyze how different outcomes interact within an experiment.

These concepts are widely used when adding probabilities or understanding combined events (like “either A or B happens”) in probability problems.

Pattern: Mutually Exclusive and Exhaustive Events

Pattern

Mutually Exclusive events cannot occur together, while Exhaustive events cover all possible outcomes.

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

Example: When tossing a coin, getting a Head and getting a Tail are mutually exclusive (cannot happen together), and together they are exhaustive because one of them must occur.

Step-by-Step Example

Question

When rolling a die, let event A = “getting an even number” and event B = “getting an odd number.” Are these events mutually exclusive and/or exhaustive?

Solution

  1. Step 1: Define outcomes

    A = {2, 4, 6} and B = {1, 3, 5}.

  2. Step 2: Check for intersection

    A ∩ B = {} → empty set, so they are mutually exclusive.

  3. Step 3: Check for exhaustiveness

    A ∪ B = {1, 2, 3, 4, 5, 6} → all possible outcomes of a die → exhaustive.

  4. Final Answer:

    mutually exclusive and exhaustive.

  5. Quick Check:

    P(A ∩ B) = 0 and P(A ∪ B) = 1 ✅

Quick Variations

1. Two cards drawn showing “red” and “black” suits → mutually exclusive but exhaustive.

2. Getting an even number vs getting a multiple of 3 on a die → not mutually exclusive (6 is common).

3. Rolling a die and checking for events A = {1,2,3}, B = {4,5,6} → exhaustive and mutually exclusive.

Trick to Always Use

  • Step 1: Check intersection - if A ∩ B = ∅, they’re mutually exclusive.
  • Step 2: Check union - if A ∪ B covers all outcomes, they’re exhaustive.
  • Step 3: For mutually exclusive events, use P(A ∪ B) = P(A) + P(B).

Summary

Summary

In the Mutually Exclusive and Exhaustive Events pattern:

  • Mutually exclusive → cannot occur together (P(A ∩ B) = 0).
  • Exhaustive → one of the events must occur (sum of probabilities = 1).
  • These concepts help simplify “either-or” probability problems.
  • Always verify both intersection and union properties to classify events correctly.

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).

Mock Test

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