At a Glance - Complementary & Mutually Exclusive Events
Complementary events are two outcomes of an event that are the only two possible outcomes.
- This is like flipping a coin and getting heads or tails. Of course, there are no other options, so these events are complementary.
- Rolling a die and getting a 1 or 2 are not complementary, since there are other outcomes that may happen (3, 4, 5, or 6).
- However, rolling a die and getting a 1 or not-a-1 are complementary (you have to roll either a 1 or not-a-1).
Mutually exclusive events are two or more outcomes of an event that cannot occur at the same time.
- Picking one card from a standard deck and choosing an ace or a king are mutually exclusive events (since you can't do both at the same time).
- However, choosing a red card or a king are not mutually exclusive (you could choose a red king).
All complementary events are mutually exclusive, but all mutually exclusive events are not necessarily complementary.
Let's note each Example as complementary, mutually exclusive, both, or neither.
Note: the piggy bank contains a Quarter (25 cents), Dime (10 cents), Nickel (5 cents), and Penny (1 cent)
Picking one coin from the piggy bank and getting a dime or a nickel.
Mutually exclusive? Yes. You can't pick a dime and a nickel at the same time
Complementary? No. There are other options, like a penny or quarter
Picking a coin from the piggy bank and getting a quarter or a coin divisible by 5.
Mutually exclusive? No
Quarters are also divisible by 5 (as well as dimes and nickels), so these are not mutually exclusive nor are they complementary.
Choosing one marble from the bag and getting a blue or a green one.
Mutually exclusive? Yes. You can't choose a marble that is both blue and green at the same time.
Complementary? No. There are other options (like an orange one).