Union: Coins

P(A ∪ B) with coin flip examples

Calculator Dice Coins Marbles Cards Learn

Head OR Specific FlipH or T At Least One H OR TAt Least Mixed OutcomesMixed

When Events Overlap: The Full Union Formula

What happens when events CAN overlap?

If events A and B can happen together (not mutually exclusive), simply adding P(A) + P(B) will double-count the overlap. We must subtract it!

The Inclusion-Exclusion Principle

To avoid double-counting: add each event, then subtract the overlap.

The General Union Formula:

Example: At Least One Head OR At Least One Tail

Question

Flip a fair coin twice. What's P(at least one head OR at least one tail)?

Sample space (all possible outcomes):

4 equally likely outcomes, each with probability ¼

Define Event A: At least one head

Which outcomes have at least one H? HH, HT, TH (3 outcomes)

Define Event B: At least one tail

Which outcomes have at least one T? HT, TH, TT (3 outcomes)

Find the overlap: A AND B (both at least one H and at least one T)

This means "mixed results": HT, TH (2 outcomes). Notice how HT and TH appear in BOTH lists above!

Apply the inclusion-exclusion formula

Without subtracting, we'd count HT and TH twice (once in A, once in B):

Result:

✓ This makes sense: every flip sequence has either a head somewhere OR a tail somewhere (or both)!

Key Insight: The Double-Counting Problem

If we just added 3/4 + 3/4 = 6/4 = 150%, we'd get an impossible probability! That's because HT and TH were counted twice. The intersection subtraction corrects this double-counting.