Union: Marbles

P(A ∪ B) with marble examples

Calculator Dice Coins Marbles Cards Learn

Red OR BlueR or B Overlapping EventsOverlap Mutually ExclusiveExclusive

When Properties Overlap

Can a marble have TWO different properties?

Yes! Unlike colors (which are mutually exclusive), other properties like "red" and "large" can overlap. A marble could be BOTH red AND large at the same time.

Why does overlap matter?

If we just add P(Red) + P(Large), marbles that are BOTH red AND large get counted twice. We must subtract the overlap to get the correct answer.

The Inclusion-Exclusion Formula:

Example: Red OR Large Marble

Question

A bag has 10 numbered marbles. Marbles 1-5 are red. Marbles 4-8 are large. What is P(Red OR Large)?

Let's visualize the sets:

Red only Both (overlap) Large only

Red: {1, 2, 3, 4, 5} — 5 marbles
Large: {4, 5, 6, 7, 8} — 5 marbles
Both (overlap): {4, 5} — 2 marbles

Add the individual probabilities

This counts marbles 4 and 5 twice!

Subtract the overlap

Remove the 2 marbles (4, 5) counted twice

Result:

✓ Verification: Marbles 1-8 satisfy "red OR large" = 8/10 ✓

Key Insight

When properties can overlap (like color AND size), you MUST use the inclusion-exclusion formula. The overlap gets counted twice if you just add, so subtract it once to correct.