ACARA v9 CONTENT DESCRIPTION “recognise that complementary events have a combined probability of one; use this relationship to calculate probabilities in applied contexts”
Probability from 0 to 1
A probability measures how likely an event is to happen, and it is always a number on a scale from 0 to 1. A probability of 0 means the event is impossible, and a probability of 1 means it is certain. A value of 0.5 sits exactly in the middle and describes an even chance, as likely to happen as not. The same probability can be written in different ways: as a fraction such as 1/2, as a decimal such as 0.5, or as a percentage such as 50%. Whichever form you use, the value never falls below 0 and never rises above 1, and that fixed scale is what makes probabilities easy to compare.
The probability scale
From impossible at 0 to certain at 1.
every probability is a number from 0 to 1, from impossible at 0 to certain at 1.
An event and its complement
The complement of an event A is simply the event that A does not happen. We write it as not A, and read it as the opposite outcome to A. If A is the event that it rains tomorrow, then not A is the event that it does not rain. The key idea is that A and not A divide every possible outcome into two groups, with nothing left over and nothing counted twice. Together they fill the whole sample space, the set of all the things that could happen, and they never overlap. Every outcome belongs to exactly one of them: either it is in A, or it is in not A.
An event and its complement
A and not A fill the whole sample space.
an event A and its complement, not A, together cover every possible outcome.
The complement rule
Because A and not A between them cover all the possibilities, their probabilities must account for everything, and so they add to 1. This is the complement rule, written P(A) + P(not A) = 1. It says that the chance of an event happening and the chance of it not happening always total one whole. Rearranging the rule gives a second, very handy form: P(not A) = 1 - P(A). In words, to find the probability that something does not happen, take its probability away from 1. These two statements are the heart of this unit, and every example below is just an application of them.
The complement rule on a bar
The two parts add to 1.
the two probabilities add to 1, so P(A) + P(not A) = 1 and P(not A) = 1 - P(A).
Working with the complement
Putting numbers into the rule makes it concrete. Suppose the weather report gives the chance of rain as P(rain) = 0.3. The chance of no rain is then P(no rain) = 1 - 0.3 = 0.7, and you can check that 0.3 + 0.7 = 1. Now roll a fair six-sided die. The probability of a six is P(six) = 1/6, so the probability of not a six is P(not six) = 1 - 1/6 = 5/6, and again 1/6 + 5/6 = 1. A good habit is to add each pair as a check: if the two probabilities do not total 1, something has gone wrong, and you can find the slip before it matters.
A fair die
P(six) = 1/6 and P(not six) = 5/6.
for a fair die, P(six) = 1/6 and P(not six) = 5/6, and 1/6 + 5/6 = 1.
When the complement is easier
Sometimes not A is far easier to count than A itself, and that is where the rule earns its keep. Take two coin tosses and ask for the chance of at least one head. Listing the ways to get at least one head takes a moment, but the opposite event, no heads at all, happens in just one way out of four, so P(no heads) = 1/4. The shortcut is to subtract: P(at least one head) = 1 - 1/4 = 3/4. This one minus the complement trick turns a fiddly count into a single subtraction. It is mentioned here only as a taste; the full treatment of such compound events comes in a later unit.
The complement as a shortcut
Count not A, then subtract from 1.
sometimes counting the complement is easier; here P(at least one head) = 1 - P(no heads) = 1 - 1/4 = 3/4.
Why this matters
The complement rule is one of the most useful tools in all of probability. Weather forecasts, games of chance, the reliability of machines, and the assessment of risk all lean on it, because the chance of something not happening is very often the quick way in to the chance that it does. The rule also reinforces a deeper point: the probabilities of a complete set of separate outcomes always add to 1, with no gaps and no overlaps. The most common slip is to forget that the two parts must sum to 1, or to miscount the complement and break that balance. Master this single idea and the rest of the year becomes much steadier. This begins the Year 8 probability units.
Quick self-check
1. What range can a probability take?
2. What is the complement of an event A?
3. What do the probabilities of an event and its complement add up to?
4. If P(rain) = 0.3, what is P(no rain)?
5. For a fair die, what is the probability of NOT rolling a six?