Complement: Marbles

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

Calculator Dice Coins Marbles Cards Learn

NOT a ColorNOT Color At Least OneAt Least Neither ColorNeither

Understanding "Neither"

What does "neither red nor blue" mean?

"Neither" means not the first AND not the second. So "neither red nor blue" means NOT red AND NOT blue — you want a completely different color.

How does complement apply here?

"Neither A nor B" is the complement of "A or B".

"Red or blue" = you get red OR blue (or both, but marbles are one color)
"Neither red nor blue" = you get something else entirely

P(neither) = 1 - P(red or blue)

Two Ways to Solve:

Direct: Count marbles that are NOT red and NOT blue, divide by total
Complement: Find P(red or blue) and subtract from 1

Both work! The direct method is often easier for "neither" problems.

The Formula:

Example: Neither Red nor Blue

Question

A bag has 3 red, 4 blue, 2 green, and 1 yellow marble. Draw one marble. What's the probability of getting neither red nor blue?

The Setup: 3 + 4 + 2 + 1 = 10 total marbles

red: 3

blue: 4

green: 2

yellow: 1

Total: 10 marbles

Identify what we DON'T want

We don't want red OR blue.

Red + Blue = 3 + 4 = 7 marbles to avoid

Count what's left (neither red nor blue)

The "other" colors: Green + Yellow = 2 + 1 = 3 marbles

Or: 10 total - 7 (red+blue) = 3 marbles

Calculate probability

3 favorable outcomes out of 10 total:

Result:

Verification: Complements Sum to 1

Let's verify our answer by checking that P(red or blue) + P(neither) = 1:

P(red or blue)

7 marbles that are red or blue out of 10:

P(neither red nor blue)

3 marbles that are neither:

Verification:

They're complements — they add up to 100%!

Key Insight

"Neither A nor B" is the complement of "A or B". For single draws with exclusive colors, you can directly count "the other colors". For more complex problems, use: P(neither) = 1 - P(A or B).