Introduction
Probability requires specific language to talk about correctly. We will now investigate some ways that two events can be related to each other.
The language of probability
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.
To determine whether or not two events are mutually exclusive, ask these questions:
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"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.
When referring to a things that include numbers we might use the terms 'at most' or 'at least'. The phrase 'at most 4', includes any number less than or equal to 4. The phrase 'at least 7', includes any number greater than or equal to 7.
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.
Examples
Example 1
Consider the following events:
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Event 1: "A six-sided die is rolled, and the result is 3 or less"
Event 2: "A six-sided die is rolled, and the result is 4 or more"
Are the events mutually exclusive?
Are the events complementary?
Mutually exclusive and complementary
To determine whether or not two events are mutually exclusive, ask these questions:
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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.