Ninth Grade Mathematics: Complementary Events: P(E) + P(not E) = 1
In probability theory, every event possesses a natural opposite: the event that the specified condition fails to happen. This opposite event is called the complementary event. The relationship between an event and its complement forms one of the most powerful computational identities in probability, enabling you to solve complex problems by simply calculating the probability of failure.
The Mathematical Definition of Complementary Events
Let E be any event defined on a sample space S. The complementary event of E (denoted by not E, E', or E-bar) is the event consisting of all outcomes in the sample space that do NOT belong to E.
Venn Diagram of Complementary Events:
+-------------------------------+
| Sample Space S |
| +---------------+ |
| | | |
| | Event E | |
| | | |
| +---------------+ |
| |
| Event NOT E |
| (E-bar) |
+-------------------------------+
S = Event E UNION Event NOT E
Event E and Event not E are mutually exclusive (they cannot both happen at the same time) and exhaustive (together, they encompass every possible outcome in the entire sample space).
The Fundamental Complementary Rule
Because the union of E and not E forms the entire sample space S, and they share no common outcomes:
n(E) + n(not E) = n(S)
Dividing every term by n(S):
n(E) n(not E) n(S)
------ + ---------- = ------
n(S) n(S) n(S)
The Fundamental Complementary Identity:
P(E) + P(not E) = 1
Equivalently:
P(not E) = 1 - P(E)
P(E) = 1 - P(not E)
The sum of the probability of an event happening and the probability of it not happening is always identically equal to 1.
Example: The probability that it will rain today is 0.35.
What is the probability that it will NOT rain today?
P(not rain) = 1 - P(rain) = 1 - 0.35 = 0.65.
The Power of the 'At Least One' Principle
Calculating the probability of an event directly can often require listing dozens of complicated combinations. However, whenever you see the phrase 'at least one', the complementary rule provides an immediate shortcut:
The 'At Least One' Identity:
The exact opposite of 'at least one' is 'none at all' (zero occurrences):
P(At least one success) = 1 - P(No successes)
Example: Toss a fair coin 3 times. What is P(Getting at least one Head)?
Direct Method:
Favorable outcomes: {HHT, HTH, HTT, THH, THT, TTH, HHH} (7 outcomes)
P(At least one Head) = 7 / 8.
Complementary Shortcut Method:
The complement of 'at least one Head' is 'ZERO Heads' (getting all Tails: TTT).
P(No Heads) = P(TTT) = 1 / 8.
P(At least one Head) = 1 - P(No Heads) = 1 - 1/8 = 7/8.
The complementary shortcut required calculating only a single outcome (TTT) instead of enumerating seven different combinations.
Mutually Exclusive Events and the Addition Rule
Two events A and B are called mutually exclusive (or disjoint) if they cannot occur simultaneously in a single trial: A intersect B = {}.
The Addition Rule for Mutually Exclusive Events:
If events A and B cannot occur together:
P(A or B) = P(A) + P(B)
If events are not mutually exclusive (meaning they can happen at the same time, such as drawing a card that is both a Red card and a King):
General Addition Rule:
P(A or B) = P(A) + P(B) - P(A and B)
Subtracting P(A and B) eliminates the double-counting of the overlapping outcomes.
Chapter Exercises
Exercise 1: If the probability of winning a game is 0.72, what is the probability of losing the game?
Exercise 2: A card is drawn from a well-shuffled deck of 52 cards. What is the probability of drawing a card that is NOT an Ace? Use the complementary rule.
Exercise 3: Two fair dice are thrown simultaneously. What is the probability of getting a doublet (both dice showing the same number)? Using the complementary rule, find the probability that the two dice show different numbers.
Solutions and Explanations
Solution 1: Winning the game and losing the game are complementary events. By the Fundamental Complementary Identity: P(Winning) + P(Losing) = 1. We are given P(Winning) = 0.72. Therefore, P(Losing) = 1 - P(Winning) = 1 - 0.72 = 0.28. The probability of losing the game is 0.28.
Solution 2: Let E be the event of drawing an Ace. In a standard 52-card deck, there are 4 Aces, so n(E) = 4 and n(S) = 52. The probability of drawing an Ace is P(E) = 4 / 52 = 1 / 13. The event of drawing a card that is NOT an Ace is the complementary event, not E. By the complementary rule: P(not E) = 1 - P(E) = 1 - 1/13 = 12/13. The probability of not drawing an Ace is 12/13.
Solution 3: When two dice are thrown simultaneously, the total number of outcomes is n(S) = 6 * 6 = 36. Let E be the event of getting a doublet (same number on both dice). The favorable outcomes are E = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}, so n(E) = 6. The probability of getting a doublet is P(E) = 6 / 36 = 1 / 6. The event of the two dice showing different numbers is the exact complement of getting a doublet (not E). By the complementary identity: P(different numbers) = 1 - P(doublet) = 1 - 1/6 = 5/6. The probability of rolling different numbers is 5/6.