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Complementary events

⭐ Complementary Events

Two events are complements when they cover the entire sample space together and cannot happen at the same time.

If A is an event, then its complement A^c is:

  • “everything in the sample space that is not in A”
  • the opposite of A
  • the rest of the universe after A is removed

📌 Key properties

P(A) + P(A^c) = 1

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

A \cap A^c = \emptyset

A \cup A^c = S

🎯 Examples

Example 1: Coin flip

Let A = “get heads.”
Then
A^c = \text{get tails.}

Example 2: Rolling a die

Let A = “roll a number greater than 4.”
A = {5,6}
A^c = {1,2,3,4}

Example 3: Real‑world

Let A = “a student is absent today.”
Then
A^c = \text{the student is present.}

🧠 Why complements matter

They make probability calculations much easier.

Sometimes it’s hard to compute P(A) directly, but very easy to compute P(A^c).
So you use:

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

This trick is used constantly in probability, especially with “at least one” problems.

🎨 A simple analogy

Think of a light switch:

On = event A
Off = event A^c

There’s no third option. Together they cover everything.

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