Two football teams, the Wollongong Warriors and the Montrose Magpies, are competing in the semi-final. Because of their success earlier in the season, the Magpies will proceed to the final if there is a draw.

There are three possible outcomes from the match - the Warriors win, the Magpies win, or there may be a draw.

The event "The Warriors win" and the event "The Magpies win" cannot both occur, so we say they are mutually exclusive.

The event "The Warriors proceed to the final" and the event "The Magpies proceed to the final" are also mutually exclusive, but unlike the previous pair, one of them must happen. We say these events are complementary.

Try thinking about these different events using the questions in the box below.

Mutually exclusive and complementary

To determine whether or not two events are mutually exclusive, ask these questions:

"If Event 1 happens, do you know for sure that Event 2 did not happen?"

"If Event 2 happens, do you know for sure that Event 1 did not happen?"

If the answer to both of these questions is "yes", the events are mutually exclusive.

To determine whether or not two events are complementary, ask these questions:

"If Event 1 does not happen, do you know for sure that Event 2 did happen?"

"If Event 2 does not happen, do you know for sure that Event 1 did happen?"

If the answer to both of these questions is "yes", the events are complementary.

Worked example

When a six-sided die is rolled, two events that could happen are "The result is at most $3$3", and "The result is $5$5 or more".

Are these events mutually exclusive? Are they complementary?

Think: Let's call "The result is at most $3$3" Event 1, and "The result is $5$5 or more" Event 2.

If Event 1 happens, the result was $1$1, $2$2, or $3$3. If it does not happen, the result was $4$4, $5$5, or $6$6.

If Event 2 happens, the result was a $5$5 or a $6$6. If it does not happen, the result was $1$1, $2$2, $3$3, or $4$4.

We will use the questions from the box above for each event.

Do: Using the questions from above,

"If Event 1 happens, do you know for sure that Event 2 did not happen?" - YES

"If Event 2 happens, do you know for sure that Event 1 did not happen?" - YES

So the events are mutually exclusive.

Now for the next set of questions, we know that the result could be a $4$4. This means

"If Event 1 does not happen, do you know for sure that Event 2 did happen?" - NO

"If Event 2 does not happen, do you know for sure that Event 1 did happen?" - NO

So the events are not complementary.

Caution

When referring to a things that include numbers we might use the terms 'at most' or 'at least'. The phrase 'at most $4$4', includes any number less than or equal to $4$4. The phrase 'at least $7$7', includes any number greater than or equal to $7$7.

Now we will examine how different events can be combined.

Exploration

Consider the $9$9 gems that are being examined by a jeweler below. Each gem can be one of three colours: ruby, emerald, or amethyst; and one of three cuts, triangular, rectangular and octagonal.

Gems examined by the jeweller

We need clear language when we want to refer to particular kinds of gems. To answer the question "How many of the gems are both emerald and octagonal cut?", we would only count a gem if it was both at the same time.

Gems that are both emerald and octagonal

To answer the question "How many are not rubies?", we would count a gem only if it was an emerald or amethyst.

Gems that are not rubies

For the question "How many are rubies or rectangular?", we would count any gem that was a ruby, rectangular, or both.

Gems that are rubies or rectangular

For the question "How many are rubies or rectangular, but not both?", we would count any gem that was a ruby, rectangular, but not include any rectangular rubies.

Gems that are rubies or rectangular but not both

Notice how in the bottom right corner the rectangular ruby is not counted.

Caution

The word "or" can mean different things in everyday language. If someone asks "Do you own a dog or a cat?", you should say "yes" if you own both. But if someone asks you "Do you want to play video games or go to the park?", they would probably be confused if you said "yes"!

In probability we will use "or" in the first way, including both. The second question will be phrased "Do you want to play video games or go to the park, but not both?" to be very clear about what we mean.