Complementary Probability

Calculating the probability that an event does not happen is often easier than calculating the probability that it does.

The Rule

Theorem (Complement Rule)

For any event

A

P (A) = 1 - P (A^{c})

where

A^{c}

(or

\overset{ˉ}{A}

) is the event that

A

does not occur.

When to Use It?

Look for keywords:

"At least one..."
"Not all..."
"At least one pair..."

Example (Birthday Problem)

In a group of

n = 23

people, what is the probability that at least two share a birthday? (Ignore leap years).

Proof. Let $A$ be the event "at least 2 share a birthday". $A^{c}$ is the event "everyone has a distinct birthday". Total ways to assign birthdays: $36 5^{23}$ . Ways to assign distinct birthdays: $365 \times 364 \times \dots \times (365 - 22) = P (365, 23)$ .

P (A^{c}) = \frac{P ( 365 , 23 )}{36 5 ^{23}} \approx 0.493

P (A) = 1 - 0.493 = 0.507

It is more likely than not! ∎

Principle of Inclusion-Exclusion (Probability Form)

For two events:

P (A \cup B) = P (A) + P (B) - P (A \cap B)

The complement is often

P ((A \cup B)^{c}) = 1 - P (A \cup B)

Practice Problems

Exercise (Problem 1)

A fair coin is flipped 10 times. What is the probability of getting at least one Head?

Exercise (Problem 2)

A die is rolled 4 times. What is the probability that the value 6 appears at least once?

Exercise (Problem 3)

Cards are drawn from a deck one by one without replacement. What is the probability that the Ace of Spades is drawn before the King of Hearts? (Hint: Symmetry/Complementary logic).