The complement of an event is everything else (in the sample space) that is not the event. If the event is that you flip tail, then the complement is head. 

Say you throw the number 6 with a standard six-sided die. We can call this “the event of throwing a 6”. The complement of this event is throwing anything else than a 6. So, the complement of throwing a 6 is throwing 1,2,3,4 or 5.

 

Notation of the complement of an event

The complement of an event can be denoted in different ways. As example let’s express the  probability of the complement of event A: 

 

 Other ways of expressing the complement rule:

A + the rest = All.

One thing + the rest = All

P(A) + P(A’) = All

 

And out of “All” equals 1 (because one = 100%) the complement rule says:

 

Complement rule

The complement of an event follows the complement rule which says “something + the rest = All”. This can be expressed: 

 

 

The complement event for throwing 1 or 2

Let A be the event that you throw 1 or 2 with a six-sided die. The complement of that event becomes, the A’, be that you throw anything else. We can denote: 

 

And visualized with a Venn diagram: 

 

Why is the complement of an event useful?

Why is it useful to calculate what we don’t want to know instead of calculating what we do want to know? Because, sometimes it’s much easier. We find the easy part first and subtract this from one, and that leaves us with what we were are looking for. Loosely speaking:

 

1 – The easy part = The difficult part

 

Example. Probability of throwing different values on 2 dice

Say you are tossing two dice. What’s the probability of each die showing different values?

Let A be the event that we throw different values. The first die could give a 1 and the second die a 3, or the first a 5 and the second a 1, etc. The complement of Event A (the Event A’) is the event that we throw two equal values: 1,1 or 2,2, etc. This is a total of 6 possible events.

So, all the rest, apart from these 6 possible events must be Event A that we are looking for. As the sample space consists of 36 possible outcomes (6×6) we find that the 6 events of throwing two dice is 6/36 = 1/6. The remaining will therefore be the result that we are looking for. We can therefore write:

 

We easily found that there was a 1/6 chance of getting what we “didn’t care about” and thus that the probability that we were looking for was the rest: the 5/6.

 

Example. The probability of flipping at least 1 tail

Say you were to flip a fair coin 8 times. What is the probability of flipping at least one tail? A coin flip has two possible outcomes: head and tail. The total number of possible outcomes for the 8 flips is 2⁸ = 256.

We see that calculating all the possibilities that satisfy our constrain of at least one tail would be more complicated than just to calculate the compliment of this event.

 The only event that does NOT satisfy our constrain is the series of 8 heads. This one and only event is the only one that does not contain a tail. Therefore, this throw has a probability of 1/256. We can now answer the question by using the complement rule: 

 

There is a 99.6% probability of flipping at least 1 tail out of 8 flips with a fair coin. We found P(A’) and subtracted this from the rest (=1).