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

Understanding "OR" in Probability

What does "OR" mean in probability?

When we say "A OR B", we mean: event A happens, event B happens, or both happen. In probability notation, this is written as P(A ∪ B) — the union of A and B.

What is "mutually exclusive"?

Two events are mutually exclusive when they can NEVER happen at the same time. A coin flip is the perfect example: you get Heads OR Tails, but never both!

Why does mutually exclusive matter?

For mutually exclusive events, there's no overlap to subtract. So the union formula simplifies from:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B) ← general formula

P(A ∪ B) = P(A) + P(B) ← when P(A ∩ B) = 0 (mutually exclusive)

For mutually exclusive events:

Example: P(Heads OR Tails) in One Flip

Question

Flip a fair coin once. What's the probability of getting Heads OR Tails?

This might seem like a trick question — you ALWAYS get one or the other! Let's prove it with the union formula.

Event A: Heads

Event B: Tails

Check: Are these mutually exclusive?

Yes! One coin flip cannot be both Heads AND Tails. So P(H ∩ T) = 0.

Apply the simplified union formula

Result:

As expected — you ALWAYS get either heads or tails. The events cover all possibilities!

Key Insight

When P(A ∪ B) = 1, the events together cover all possible outcomes. This happens when H and T are the ONLY two possibilities, and they're mutually exclusive. Such events are called collectively exhaustive.