Tenth Grade Mathematics: Complementary Events and Mutually Exclusive Events

In formal probability theory, the relationship between events is analyzed using the algebraic operations of set theory: union, intersection, and complementation. When two events cannot occur simultaneously on a single trial, they are mutually exclusive. When an event represents the exact logical negation of another—occurring if and only if the other fails to occur—the events are complementary. Mastering these set-theoretic relationships enables statisticians to solve complex multi-stage problems through the Law of Total Probability and the Addition Theorem.

The Concept of Complementary Events

For any event E defined on a sample space S, the event "not E" is called the complement of E, denoted by E' (or E_bar, or not-E):

Set-Theoretic Definition:
The complement E' consists of all sample points in S that DO NOT belong to E:
                             E' = S - E

Visualizing Complementary Events via Venn Diagram:
+---------------------------------------------+
| Sample Space S                              |
|          +--------------------+             |
|          |                    |             |
|          |      Event E       |   Event E'  |
|          |                    |   (Not E)   |
|          +--------------------+             |
|                                             |
+---------------------------------------------+

The Fundamental Complementary Properties:
1. Union covers the entire sample space:   E U E' = S
2. Intersection is empty (disjoint):       E intersect E' = phi

The Complementary Probability Rule

Since E and E' partition the sample space S into two non-overlapping subsets: P(E U E') = P(S) = 1. Because they share no common elements: P(E U E') = P(E) + P(E').

Theorem (The Complementary Probability Formula):
                         P(E) + P(E') = 1
                       P(E') = 1 - P(E)

The Power of the Complement Rule: "At Least One" Problems:
In complex multi-item probability questions, computing the probability of
"at least one success" directly requires summing multiple cases.
Using the complement rule makes the computation instantaneous:
             P(At least one success) = 1 - P(Zero successes)

Mutually Exclusive Events

Two events A and B are called mutually exclusive (or disjoint) if they cannot both happen at the exact same time:

Formal Definition:
Two events A and B are mutually exclusive if their intersection is the empty set:
                         A intersect B = phi
                         P(A intersect B) = 0

Examples:
1. Rolling a single die:
   Event A = Getting an odd number {1, 3, 5}
   Event B = Getting an even number {2, 4, 6}
   A intersect B = phi  ===>  A and B are MUTUALLY EXCLUSIVE!

2. Drawing a card from a deck:
   Event A = Drawing a King
   Event B = Drawing a Heart
   Can a card be both a King and a Heart? YES (The King of Hearts!).
   A intersect B = {King of Hearts} != phi  ===>  NOT mutually exclusive!

The Addition Theorem of Probability

The probability that at least one of two events A or B occurs—corresponding to their union A U B—is governed by the Principle of Inclusion-Exclusion:

Theorem (General Addition Theorem):
For any two arbitrary events A and B:
                 P(A U B) = P(A) + P(B) - P(A intersect B)

When we add P(A) and P(B), the outcomes belonging to their intersection (A intersect B) are counted twice. Subtracting P(A intersect B) corrects this double-counting.

Corollary for Mutually Exclusive Events:
If A and B are mutually exclusive, P(A intersect B) = 0:
                 P(A U B) = P(A) + P(B)

Exhaustive Events

A set of events E1, E2, ..., Ek is called exhaustive if their union covers the entire sample space: E1 U E2 U ... U Ek = S. If events are both mutually exclusive and exhaustive, they form a complete partition of the sample space, and their probabilities sum to 1.

Chapter Exercises

Exercise 1: If P(E) = 0.05, what is the probability of 'not E'?

Exercise 2: A card is drawn at random from a well-shuffled pack of 52 cards. Let Event A be drawing a red card, and Event B be drawing a King. Determine whether A and B are mutually exclusive, and find P(A U B) using the Addition Theorem.

Exercise 3: Two players, Sangeeta and Reshma, play a tennis match. The probability of Sangeeta winning the match is 0.62. What is the probability of Reshma winning the match?

Solutions and Explanations

Solution 1: We are given the probability of event E as P(E) = 0.05. By the Complementary Probability Formula: P(not E) = 1 - P(E). Substituting the value: P(not E) = 1 - 0.05 = 0.95. Thus, the probability of 'not E' is 0.95.

Solution 2: In a standard deck of 52 cards, total outcomes n(S) = 52. Event A is drawing a red card; there are 26 red cards (13 hearts + 13 diamonds), so P(A) = 26/52 = 1/2. Event B is drawing a King; there are 4 Kings in the deck, so P(B) = 4/52 = 1/13. The intersection A intersect B represents drawing a card that is both red AND a King; there are 2 red Kings (King of Hearts and King of Diamonds). Since A intersect B != phi, Events A and B are NOT mutually exclusive. P(A intersect B) = 2/52 = 1/26. Applying the Addition Theorem: P(A U B) = P(A) + P(B) - P(A intersect B) = 26/52 + 4/52 - 2/52 = (26 + 4 - 2) / 52 = 28 / 52. Dividing numerator and denominator by 4: P(A U B) = 7 / 13.

Solution 3: In a competitive tennis match between two players, a draw is not possible; one player must win and the other must lose. Therefore, the event of Reshma winning the match is identically the complementary event of Sangeeta winning the match: Event(Reshma wins) = not(Sangeeta wins). Using the complement rule: P(Reshma wins) = 1 - P(Sangeeta wins) = 1 - 0.62 = 0.38. Thus, the probability of Reshma winning the match is 0.38.