Complementary Events

Example: Rolling a Die

A coin toss (Heads/Tails) is the simplest example, but rolling a standard 6-sided die illustrates the concept better because it shows how the “Complement” can include multiple outcomes.

Imagine you want to roll a 5.

Event A (Rolling a 5): There is only one face with a 5.
\(P(A) = \frac{1}{6}\)

Event A' (Not rolling a 5): This includes rolling a 1, 2, 3, 4, or 6.
\(P(A') = \frac{5}{6}\)

The Check: When you add them together, they equal the whole (100%):
\(\frac{1}{6} + \frac{5}{6} = \frac{6}{6} = 1\)
 

In complementary events, the sum of probabilities is always equal to 1. If the probability of event A is P(A), the probability of event A^c is P(A^c) = 1 - P(A). We call this the probability sum rule. 

Mathematically, it is expressed as:

P(A’) = 1 - P(A)

P(A) = 1 - P(A’)

P(A) + P(A’) = 1

These three statements are all equivalent.

Where:

P(A’): Is the probability of complementary event A’.

P(A): Is the Probability of event A.
 

Complementary events are defined by three strict rules. If two events do not meet all of these criteria, they are not complements.

1. They are Mutually Exclusive
The two events generally cannot happen at the same time. There is no overlap between them.

• Math: \(P(A \cap A') = 0\)
   
• Example: You cannot roll a die and get a 5 and "Not a 5" simultaneously. It is impossible.

2. They are Exhaustive
Together, the two events cover every possible outcome in the sample space. There is no third option left out.

• Math: \(A \cup A' = S \)(The Sample Space)

• Example: When you roll a die, the result must be either “5” or “Not 5.” There is no other magic number that falls outside these two categories.

3. Their Probabilities Sum to 1
Because they cover everything (exhaustiveness) and don't overlap (mutual exclusivity), their probabilities must add up to 100%.

• Math: \(P(A) + P(A') = 1\)

• Utility: This is the most useful property because it allows you to use subtraction to solve hard problems:
  \(P(A) = 1 - P(A')\)

Complementary events is a complex topic, and in this section we will discuss some tips and tricks that can help us master complementary events.

• The sum of probabilities for an event and its complement always equals 1 (or 100%), representing the entire “pie” or sample space.
  
   
• Calculating the complement is often a faster “shortcut” for solving complex problems by finding what didn't happen and subtracting it from the total.
  
   
• The keyword “NOT” in a question serves as an immediate indicator to calculate the event's probability and subtract it from 1.

• The phrase “at least one” is the specific trigger for using complements, as calculating the probability of “none” is mathematically simpler than calculating “one or more.”
  
   
• Complementary events are strictly mutually exclusive, meaning they can never occur simultaneously; an outcome is either inside the target group or outside of it, with no overlap.
  
   
• The existence of two complementary options does not imply equal odds; an event either happening or not happening does not guarantee a 50/50 chance (e.g., winning the lottery vs. not winning).

Complementary events are events that occur only if the other event does not occur. Here are a few real-world applications of complementary events.

 

• Medical testing: When testing a patient for a disease, they would either have the disease or not. They will not be able to have both. 
   
• Sports outcomes: In a knockout match with no draws, a team will either win or lose.
   
• Airline safety: A flight will either arrive on time or will be delayed.
   
• Quality control in manufacturing: A product in a factory is either defective or not defective. Both cannot happen simultaneously. This helps in calculating defect rates and planning inspections.
   
• Exam results: A student either passes or fails an exam. These are complementary events, useful in analyzing pass/fail probabilities.