Complement: Dice

P(NOT E) = 1 - P(E) with dice examples

Natural Complement Pairs

What are Natural Complements?

Some events are natural opposites — if one doesn't happen, the other MUST happen. There's no third option.

Examples of natural complement pairs:

• Even vs Odd: Every whole number is either even or odd, never both, never neither
• Greater than 3 vs Less than or equal to 3: A number must be one or the other
• Heads vs Tails: A coin flip must be one or the other

Question

What is the probability of NOT rolling an even number?

✓

3 even numbers out of 6 total =

✓

3 odd numbers out of 6 total =

Why does this make sense?

• A die has 6 faces: 1, 2, 3, 4, 5, 6
• Half of these (2, 4, 6) are even
• The other half (1, 3, 5) are odd
• So even and odd each have 50% probability — they perfectly split the possibilities!

Verification that complements add to 1:

This must always be true for complements!

Question

What is the probability of rolling 4 or less?

First, understand what we're asking:

"NOT greater than 4" means the number is 4 or less. In math notation: ≤ 4. So we want: 1, 2, 3, or 4.

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2 numbers out of 6:

✓

4 numbers out of 6:

Using the complement rule:

We know P(> 4) = 1/3, so:

This matches our direct count of 4/6! ✓

Notice how each pair adds up to 1 (100%). This is always true for complement pairs!

Event	P(E)	Complement (NOT E)	P(NOT E)
Even (2,4,6)		Odd (1,3,5)
> 3 (4,5,6)		≤ 3 (1,2,3)
> 4 (5,6)		≤ 4 (1,2,3,4)
Prime (2,3,5)		Not Prime (1,4,6)