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, also called disjoint 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.
Example 1
Using the images below, determine if each event is 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). a. Picking one coin from the piggy bank and getting a dime or a nickel. b. Picking one coin from the piggy bank and getting a quarter or a coin divisible by 5. c. Choosing one marble from the bag and getting a blue or a green one. |
Example 2
A bag has 8 neon green gumballs and 12 yellow gumballs. What is the probability of randomly picking out a neon green gumball, and is this event complementary?
|
or 40%. Example 3
The probability of pulling a blue marble randomly from the bag is |
or 
. What is the probability of picking a marble that is not blue?
, the probability of picking a marble that is not blue is 
or 
.