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

Union of "Exactly n" Events

Can "exactly 2 heads" and "exactly 2 tails" overlap?

Think about it: if you flip 3 coins and get exactly 2 heads (like HHT), you automatically have exactly 1 tail. You can't have exactly 2 of BOTH!

Conclusion: These events are mutually exclusive. No overlap to worry about.

Why does this matter?

Since "exactly 2 heads" and "exactly 2 tails" can't happen together, we can simply add their probabilities. No subtraction needed!

Example: Exactly 2 Heads OR Exactly 2 Tails

Question

Flip a fair coin 3 times. What's P(exactly 2 heads OR exactly 2 tails)?

Sample space (2³ = 8 outcomes):

HHH

HHT

HTH

HTT

THH

THT

TTH

TTT

Exactly 2 heads Exactly 2 tails

Count outcomes with exactly 2 heads

HHT, HTH, THH = 3 outcomes

Count outcomes with exactly 2 tails

HTT, THT, TTH = 3 outcomes

Check: Is there any overlap?

No! The sets {HHT, HTH, THH} and {HTT, THT, TTH} have no common elements. P(A ∩ B) = 0 (mutually exclusive).

Add probabilities (no overlap to subtract)

Result:

✓ Verification: 6 highlighted outcomes out of 8 total = 6/8 = 75% ✓

Key Insight

"Exactly n" events are often mutually exclusive because you can't have two different exact counts simultaneously. When you see "exactly X OR exactly Y", first check if they can overlap. If not, just add the probabilities!